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   "source": [
    "from sage.schemes.riemann_surfaces.riemann_surface import RiemannSurface\n",
    "R.<z,w>=QQ[]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Computing period matrices\n",
    "Nils Bruin (SFU), August 1, 2017, Leiden\n",
    "\n",
    "Based on joint work with Alexandre Zotine (SFU)"
   ]
  },
  {
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     "slide_type": "slide"
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   "source": [
    "## 1. Complete algebraic Riemann surfaces\n",
    "\n",
    "The algebraic way of getting them:\n",
    " 1. Take an affine plane algebraic curve \n",
    "$$C^\\mathrm{aff}\\colon f(z,w)=0 \\text{ with } f(z,w)\\in \\mathbb{C}[z,w].$$\n",
    " 2. Take the smooth projective curve $C$ it determines\n",
    " 3. Consider $C(\\mathbb{C})$.\n",
    "\n",
    "**Note:** By considering $f(z,w)\\in \\mathbb{C}(z)[w]$, we get $C=C(\\mathbb{C})$ as a (ramified) finite cover of $\\mathbb{P}^1(\\mathbb{C})$."
   ]
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   "source": [
    "Several relevant quantities:\n",
    " 1. Genus $g$ of $C$\n",
    " 2. $H_1(C,\\mathbb{Z})\\simeq \\mathbb{Z}^{2g}$: Homology classes of paths on $C$\n",
    " 3. $H^0(C,\\Omega^1_C)\\simeq \\mathbb{C}^g$: Holomorphic differentials on $C$\n",
    " \n",
    "**Integration pairing:** Given $\\omega\\in H^0(C,\\Omega^1_C)$ and $[\\gamma]\\in H_1(C,\\mathbb{Z})$ we can compute\n",
    " $$\\int_\\gamma \\omega$$\n",
    " \n",
    "**Given:**\n",
    " 1. Basis $\\omega_1,\\ldots,\\omega_g$ for $H^0(C,\\Omega^1_C)$\n",
    " 2. Basis $[\\gamma_1],\\ldots,[\\gamma_{2g}]$ for $H_1(C,\\mathbb{Z})$\n",
    "\n",
    "**Period matrix:** $\\mathbb{C}$-valued $g\\times 2g$-dimensional matrix:\n",
    "$$\\begin{pmatrix}\n",
    "\\int_{\\gamma_1}\\omega_1&\\int_{\\gamma_2}\\omega_1&\\cdots&\\int_{\\gamma_{2g}}\\omega_1\\\\\n",
    "\\int_{\\gamma_1}\\omega_2&\\int_{\\gamma_2}\\omega_2&\\cdots&\\int_{\\gamma_{2g}}\\omega_2\\\\\n",
    "\\vdots&&\\ddots&\\vdots\\\\\n",
    "\\int_{\\gamma_1}\\omega_g&\\int_{\\gamma_2}\\omega_g&\\cdots&\\int_{\\gamma_{2g}}\\omega_g\\\\\n",
    "\\end{pmatrix}$$\n",
    "* The $\\mathbb{Z}$-span of the columns gives a lattice $\\Lambda\\subset \\mathbb{C}^g$.\n",
    "\n",
    "* Analytic description of the Jacobian of $C$: $\\mathbb{C}^g/\\Lambda$.\n",
    "\n",
    "* In this definition you can already see that the period matrix is only well-defined up to multiplication by $\\mathrm{GL_{2g}}(\\mathbb{Z})$ to the right and $\\mathrm{GL}_{g}(\\mathbb{C})$ to the left."
   ]
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   "source": [
    "### Outline for an algorithm to approximate a period matrix:\n",
    " 1. Determine a homology basis\n",
    " 2. Determine a basis for holomorphic differentials (purely algebraic-geometric question)\n",
    " 3. Numerically integrate the path integrals"
   ]
  },
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   "source": [
    "## 2. Computing a homology basis\n",
    "\n",
    "We consider the affine model\n",
    "$$f(z,w)=0 \\text{ where } f\\in\\mathbb{C}[z,w]$$\n",
    "as a finite cover of the $z$-plane.\n",
    "\n",
    "**Critical locus:**\n",
    "$$B=\\{ b\\in \\mathbb{C} : \\mathrm{res}_w(f,\\frac{\\partial f}{\\partial w})(b)=0\\}$$\n",
    "\n",
    "Let $d=\\mathrm{deg}_w(f)$. Then the open part of $C$ covering $\\mathbb{P}^1(\\mathbb{C})\\setminus (B\\cup\\{\\infty\\})$ is an unramified degree $d$ cover.\n",
    "\n",
    "**Note:** $B$ also includes $z$-values of singularities of our model of $C$, so some critical points might not be branch points for $z\\colon C\\to\\mathbb{P}^1$."
   ]
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   "source": [
    "**Example:** $f(z,w)=w^2-z^4-z$"
   ]
  },
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   ],
   "source": [
    "R.<z,w>=QQ[]\n",
    "f=w^2-z^4-z\n",
    "S=RiemannSurface(f)\n",
    "points(S.branch_locus,size=40,aspect_ratio=1,figsize=4)"
   ]
  },
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   "source": [
    "**Observations:**\n",
    " 1. The fundamental group of $\\mathbb{C}\\setminus B$ is generated by taking loops around each $b\\in B$.\n",
    " 2. If we lift such loops to $C$ and decompose, we get generators for the fundamental group of $C$.\n",
    " 3. We will need to analytically continue the algebraic function $w$ (which, when regarded as a \"meromorphic function\" on $\\mathbb{C}$ is multi-valued) along these paths, which will be harder near branch points.\n",
    "\n",
    "**Solution:** Build loops from edges of the Voronoi decomposition of $\\mathbb{C}$ with respect to $B$. Add some extra points far away to ensure that cells around points in $B$ are finite."
   ]
  },
  {
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DjgQB4PTp09jZ2eHo6EhgYKDaT4iKYiwmkiRx8+bNbCd5/vnnHwAqVqyIhYUF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\n",
      "text/plain": [
       "Graphics object consisting of 29 graphics primitives"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "points(S.branch_locus,axes=false,size=30,figsize=5)+S.downstairs_graph().plot()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Lifting the edges:\n",
    " 1. Above each vertex $i$ with $z=z_i$, we determine the $d$ roots $w_{i,1},\\ldots,w_{i,d}$ of $f(z_i,w)=0$\n",
    " 2. For each edge $(i,j)$ in the Voronoi decomposition, we analytically continue the algebraic function $w(z)$ with $w(z_i)=w_{i,k}$ along the edge to find $w_{z_j}=w_{j,\\sigma(k)}$, for some permutation $\\sigma=\\sigma_{i,j}\\in S_d$.\n",
    " 3. We get a graph with $d$ times the number of vertices: $(i,1),\\ldots,(i,d)$ and $d$ times the number of edges: $(i,k)\\rightarrow(j,\\sigma_{i,j}(k))$.\n",
    "\n",
    "This lifted graph on $C$ contains a homology basis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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TvPvhv2yNWls7e1t0dHSIjo4u0nwvQzgfQYFs3ryZvLw8+vbtK5sN+aVFbt28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\nRORyDB8iInK5/wNRFT6aRj9QPgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "Graphics object consisting of 55 graphics primitives"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G=Graph(S.upstairs_edges())\n",
    "G.plot(vertex_size=900,figsize=7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<iframe srcdoc=\"<!DOCTYPE html>\n",
       "<html>\n",
       "<head>\n",
       "<title></title>\n",
       "<meta charset=&quot;utf-8&quot;>\n",
       "<meta name=viewport content=&quot;width=device-width, user-scalable=no, minimum-scale=1.0, maximum-scale=1.0&quot;>\n",
       "<style>\n",
       "\n",
       "    body { margin: 0px; overflow: hidden; }\n",
       "  \n",
       "</style>\n",
       "</head>\n",
       "\n",
       "<body>\n",
       "\n",
       "<script src=&quot;/nbextensions/threejs/three.min.js&quot;></script>\n",
       "<script src=&quot;/nbextensions/threejs/OrbitControls.js&quot;></script>\n",
       "<script>\n",
       "  if ( !window.THREE ) document.write('\n",
       "<script src=&quot;https://cdn.rawgit.com/mrdoob/three.js/r80/build/three.min.js&quot;><\\/script>\n",
       "<script src=&quot;https://cdn.rawgit.com/mrdoob/three.js/r80/examples/js/controls/OrbitControls.js&quot;><\\/script>\n",
       "            ');\n",
       "</script>\n",
       "        \n",
       "<script>\n",
       "\n",
       "    var scene = new THREE.Scene();\n",
       "\n",
       "    var renderer = new THREE.WebGLRenderer( { antialias: true } );\n",
       "    renderer.setSize( window.innerWidth, window.innerHeight );\n",
       "    renderer.setClearColor( 0xffffff, 1 );\n",
       "    document.body.appendChild( renderer.domElement );\n",
       "\n",
       "    var options = {&quot;aspect_ratio&quot;: [1.0, 1.0, 1.0], &quot;decimals&quot;: 2, &quot;frame&quot;: false, &quot;axes&quot;: false, &quot;axes_labels&quot;: false};\n",
       "\n",
       "    // When animations are supported by the viewer, the value 'false'\n",
       "    // will be replaced with an option set in Python by the user\n",
       "    var animate = false; // options.animate;\n",
       "\n",
       "    var b = [{&quot;x&quot;:0.277156551846, &quot;y&quot;:-0.824312368852, &quot;z&quot;:-0.674801748471}, {&quot;x&quot;:2.04887958276, &quot;y&quot;:0.963913361282, &quot;z&quot;:0.935320496475}]; // bounds\n",
       "\n",
       "    if ( b[0].x === b[1].x ) {\n",
       "        b[0].x -= 1;\n",
       "        b[1].x += 1;\n",
       "    }\n",
       "    if ( b[0].y === b[1].y ) {\n",
       "        b[0].y -= 1;\n",
       "        b[1].y += 1;\n",
       "    }\n",
       "    if ( b[0].z === b[1].z ) {\n",
       "        b[0].z -= 1;\n",
       "        b[1].z += 1;\n",
       "    }\n",
       "\n",
       "    var rRange = Math.sqrt( Math.pow( b[1].x - b[0].x, 2 )\n",
       "                            + Math.pow( b[1].x - b[0].x, 2 ) );\n",
       "    var xRange = b[1].x - b[0].x;\n",
       "    var yRange = b[1].y - b[0].y;\n",
       "    var zRange = b[1].z - b[0].z;\n",
       "\n",
       "    var ar = options.aspect_ratio;\n",
       "    var a = [ ar[0], ar[1], ar[2] ]; // aspect multipliers\n",
       "    var autoAspect = 2.5;\n",
       "    if ( zRange > autoAspect * rRange && a[2] === 1 ) a[2] = autoAspect * rRange / zRange;\n",
       "\n",
       "    var xMid = ( b[0].x + b[1].x ) / 2;\n",
       "    var yMid = ( b[0].y + b[1].y ) / 2;\n",
       "    var zMid = ( b[0].z + b[1].z ) / 2;\n",
       "\n",
       "    var box = new THREE.Geometry();\n",
       "    box.vertices.push( new THREE.Vector3( a[0]*b[0].x, a[1]*b[0].y, a[2]*b[0].z ) );\n",
       "    box.vertices.push( new THREE.Vector3( a[0]*b[1].x, a[1]*b[1].y, a[2]*b[1].z ) );\n",
       "    var boxMesh = new THREE.LineSegments( box );\n",
       "    if ( options.frame ) scene.add( new THREE.BoxHelper( boxMesh, 'black' ) );\n",
       "\n",
       "    if ( options.axes_labels ) {\n",
       "        var d = options.decimals; // decimals\n",
       "        var offsetRatio = 0.1;\n",
       "        var al = options.axes_labels;\n",
       "\n",
       "        var offset = offsetRatio * a[1]*( b[1].y - b[0].y );\n",
       "        var xm = xMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(xm) ) xm = xm.substr(1);\n",
       "        addLabel( al[0] + '=' + xm, a[0]*xMid, a[1]*b[1].y+offset, a[2]*b[0].z );\n",
       "        addLabel( ( b[0].x ).toFixed(d), a[0]*b[0].x, a[1]*b[1].y+offset, a[2]*b[0].z );\n",
       "        addLabel( ( b[1].x ).toFixed(d), a[0]*b[1].x, a[1]*b[1].y+offset, a[2]*b[0].z );\n",
       "\n",
       "        var offset = offsetRatio * a[0]*( b[1].x - b[0].x );\n",
       "        var ym = yMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(ym) ) ym = ym.substr(1);\n",
       "        addLabel( al[1] + '=' + ym, a[0]*b[1].x+offset, a[1]*yMid, a[2]*b[0].z );\n",
       "        addLabel( ( b[0].y ).toFixed(d), a[0]*b[1].x+offset, a[1]*b[0].y, a[2]*b[0].z );\n",
       "        addLabel( ( b[1].y ).toFixed(d), a[0]*b[1].x+offset, a[1]*b[1].y, a[2]*b[0].z );\n",
       "\n",
       "        var offset = offsetRatio * a[1]*( b[1].y - b[0].y );\n",
       "        var zm = zMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(zm) ) zm = zm.substr(1);\n",
       "        addLabel( al[2] + '=' + zm, a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*zMid );\n",
       "        addLabel( ( b[0].z ).toFixed(d), a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*b[0].z );\n",
       "        addLabel( ( b[1].z ).toFixed(d), a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*b[1].z );\n",
       "    }\n",
       "\n",
       "    function addLabel( text, x, y, z ) {\n",
       "        var fontsize = 14;\n",
       "\n",
       "        var canvas = document.createElement( 'canvas' );\n",
       "        canvas.width = 128;\n",
       "        canvas.height = 32; // powers of two\n",
       "\n",
       "        var context = canvas.getContext( '2d' );\n",
       "        context.fillStyle = 'black';\n",
       "        context.font = fontsize + 'px monospace';\n",
       "        context.textAlign = 'center';\n",
       "        context.textBaseline = 'middle';\n",
       "        context.fillText( text, .5*canvas.width, .5*canvas.height );\n",
       "\n",
       "        var texture = new THREE.Texture( canvas );\n",
       "        texture.needsUpdate = true;\n",
       "\n",
       "        var sprite = new THREE.Sprite( new THREE.SpriteMaterial( { map: texture } ) );\n",
       "        sprite.position.set( x, y, z );\n",
       "        sprite.scale.set( 1, .25 ); // ratio of width to height\n",
       "        scene.add( sprite );\n",
       "    }\n",
       "\n",
       "    if ( options.axes ) scene.add( new THREE.AxisHelper( Math.min( a[0]*b[1].x, a[1]*b[1].y, a[2]*b[1].z ) ) );\n",
       "\n",
       "    var camera = new THREE.PerspectiveCamera( 45, window.innerWidth / window.innerHeight, 0.1, 1000 );\n",
       "    camera.up.set( 0, 0, 1 );\n",
       "    camera.position.set( a[0]*(xMid+xRange), a[1]*(yMid+yRange), a[2]*(zMid+zRange) );\n",
       "\n",
       "    var lights = [{&quot;x&quot;:-5, &quot;y&quot;:3, &quot;z&quot;:0, &quot;color&quot;:&quot;#7f7f7f&quot;, &quot;parent&quot;:&quot;camera&quot;}];\n",
       "    for ( var i=0 ; i < lights.length ; i++ ) {\n",
       "        var light = new THREE.DirectionalLight( lights[i].color, 1 );\n",
       "        light.position.set( a[0]*lights[i].x, a[1]*lights[i].y, a[2]*lights[i].z );\n",
       "        if ( lights[i].parent === 'camera' ) {\n",
       "            light.target.position.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "            scene.add( light.target );\n",
       "            camera.add( light );\n",
       "        } else scene.add( light );\n",
       "    }\n",
       "    scene.add( camera );\n",
       "\n",
       "    var ambient = {&quot;color&quot;:&quot;#7f7f7f&quot;};\n",
       "    scene.add( new THREE.AmbientLight( ambient.color, 1 ) );\n",
       "\n",
       "    var controls = new THREE.OrbitControls( camera, renderer.domElement );\n",
       "    controls.target.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "    controls.addEventListener( 'change', function() { if ( !animate ) render(); } );\n",
       "\n",
       "    window.addEventListener( 'resize', function() {\n",
       "        \n",
       "        renderer.setSize( window.innerWidth, window.innerHeight );\n",
       "        camera.aspect = window.innerWidth / window.innerHeight;\n",
       "        camera.updateProjectionMatrix();\n",
       "        if ( !animate ) render();\n",
       "        \n",
       "    } );\n",
       "\n",
       "    var texts = [];\n",
       "    for ( var i=0 ; i < texts.length ; i++ )\n",
       "        addLabel( texts[i].text, a[0]*texts[i].x, a[1]*texts[i].y, a[2]*texts[i].z );\n",
       "\n",
       "    var points = [];\n",
       "    for ( var i=0 ; i < points.length ; i++ ) addPoint( points[i] );\n",
       "\n",
       "    function addPoint( json ) {\n",
       "        var geometry = new THREE.Geometry();\n",
       "        var v = json.point;\n",
       "        geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "\n",
       "        var canvas = document.createElement( 'canvas' );\n",
       "        canvas.width = 128;\n",
       "        canvas.height = 128;\n",
       "\n",
       "        var context = canvas.getContext( '2d' );\n",
       "        context.arc( 64, 64, 64, 0, 2 * Math.PI );\n",
       "        context.fillStyle = json.color;\n",
       "        context.fill();\n",
       "\n",
       "        var texture = new THREE.Texture( canvas );\n",
       "        texture.needsUpdate = true;\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var material = new THREE.PointsMaterial( { size: json.size/100, map: texture,\n",
       "                                                   transparent: transparent, opacity: json.opacity,\n",
       "                                                   alphaTest: .1 } );\n",
       "\n",
       "        var c = geometry.center().multiplyScalar( -1 );\n",
       "        var mesh = new THREE.Points( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        scene.add( mesh );\n",
       "    }\n",
       "\n",
       "    var lines = [];\n",
       "    for ( var i=0 ; i < lines.length ; i++ ) addLine( lines[i] );\n",
       "\n",
       "    function addLine( json ) {\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i < json.points.length - 1 ; i++ ) {\n",
       "            var v = json.points[i];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "            var v = json.points[i+1];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "        }\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var material = new THREE.LineBasicMaterial( { color: json.color, linewidth: json.linewidth,\n",
       "                                                      transparent: transparent, opacity: json.opacity } );\n",
       "\n",
       "        var c = geometry.center().multiplyScalar( -1 );\n",
       "        var mesh = new THREE.LineSegments( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        scene.add( mesh );\n",
       "    }\n",
       "\n",
       "    var surfaces = 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&quot;opacity&quot;:1},{&quot;vertices&quot;:[{&quot;x&quot;:1.69269,&quot;y&quot;:-0.406241,&quot;z&quot;:-0.0883139},{&quot;x&quot;:1.68646,&quot;y&quot;:-0.426227,&quot;z&quot;:-0.0776086},{&quot;x&quot;:1.69973,&quot;y&quot;:-0.443573,&quot;z&quot;:-0.0863157},{&quot;x&quot;:1.71416,&quot;y&quot;:-0.434308,&quot;z&quot;:-0.102402},{&quot;x&quot;:1.70981,&quot;y&quot;:-0.411236,&quot;z&quot;:-0.103637},{&quot;x&quot;:1.981,&quot;y&quot;:-0.335837,&quot;z&quot;:0.210821},{&quot;x&quot;:1.97477,&quot;y&quot;:-0.355822,&quot;z&quot;:0.221526},{&quot;x&quot;:1.98804,&quot;y&quot;:-0.373168,&quot;z&quot;:0.212819},{&quot;x&quot;:2.00247,&quot;y&quot;:-0.363903,&quot;z&quot;:0.196732},{&quot;x&quot;:1.99812,&quot;y&quot;:-0.340831,&quot;z&quot;:0.195497}], &quot;faces&quot;:[[0,1,6,5],[1,2,7,6],[2,3,8,7],[3,4,9,8],[4,0,5,9]], &quot;color&quot;:&quot;#000000&quot;, &quot;opacity&quot;:1},{&quot;vertices&quot;:[{&quot;x&quot;:0.919164,&quot;y&quot;:-0.153559,&quot;z&quot;:0.557482},{&quot;x&quot;:0.912767,&quot;y&quot;:-0.136714,&quot;z&quot;:0.572585},{&quot;x&quot;:0.891465,&quot;y&quot;:-0.142141,&quot;z&quot;:0.580926},{&quot;x&quot;:0.884697,&quot;y&quot;:-0.16234,&quot;z&quot;:0.570977},{&quot;x&quot;:0.901816,&quot;y&quot;:-0.169397,&quot;z&quot;:0.556488},{&quot;x&quot;:1.09306,&quot;y&quot;:-0.363349,&quot;z&quot;:0.865111},{&quot;x&quot;:1.08667,&quot;y&quot;:-0.346504,&quot;z&quot;:0.880214},{&quot;x&quot;:1.06536,&quot;y&quot;:-0.351931,&quot;z&quot;:0.888555},{&quot;x&quot;:1.0586,&quot;y&quot;:-0.37213,&quot;z&quot;:0.878606},{&quot;x&quot;:1.07571,&quot;y&quot;:-0.379187,&quot;z&quot;:0.864117}], &quot;faces&quot;:[[0,1,6,5],[1,2,7,6],[2,3,8,7],[3,4,9,8],[4,0,5,9]], &quot;color&quot;:&quot;#000000&quot;, &quot;opacity&quot;:1}];\n",
       "    for ( var i=0 ; i < surfaces.length ; i++ ) addSurface( surfaces[i] );\n",
       "\n",
       "    function addSurface( json ) {\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i < json.vertices.length ; i++ ) {\n",
       "            var v = json.vertices[i];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v.x, a[1]*v.y, a[2]*v.z ) );\n",
       "        }\n",
       "        for ( var i=0 ; i < json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j < f.length - 2 ; j++ ) {\n",
       "                geometry.faces.push( new THREE.Face3( f[0], f[j+1], f[j+2] ) );\n",
       "            }\n",
       "        }\n",
       "        geometry.computeVertexNormals();\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var material = new THREE.MeshPhongMaterial( {\n",
       "                                     color: json.color, side: THREE.DoubleSide,\n",
       "                                     transparent: transparent, opacity: json.opacity,\n",
       "                                     shininess: 20 } );\n",
       "\n",
       "        var c = geometry.center().multiplyScalar( -1 );\n",
       "        var mesh = new THREE.Mesh( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        scene.add( mesh );\n",
       "    }\n",
       "\n",
       "    var scratch = new THREE.Vector3();\n",
       "\n",
       "    function render() {\n",
       "\n",
       "        if ( animate ) requestAnimationFrame( render );\n",
       "        renderer.render( scene, camera );\n",
       "\n",
       "        for ( var i=0 ; i < scene.children.length ; i++ ) {\n",
       "            if ( scene.children[i].type === 'Sprite' ) {\n",
       "                var sprite = scene.children[i];\n",
       "                var adjust = scratch.addVectors( sprite.position, scene.position )\n",
       "                               .sub( camera.position ).length() / 5;\n",
       "                sprite.scale.set( adjust, .25*adjust ); // ratio of canvas width to height\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "    \n",
       "    render();\n",
       "    controls.update();\n",
       "    if ( !animate ) render();\n",
       "\n",
       "</script>\n",
       "\n",
       "</body>\n",
       "</html>\" \n",
       "        width=\"100%\"\n",
       "        height=\"400\"\n",
       "        style=\"border: 0;\">\n",
       "</iframe>\n"
      ],
      "text/plain": [
       "Graphics3d Object"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G.plot3d(viewer=\"threejs\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
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   "source": [
    "### Strategy for analytic continuation\n",
    "* Given values $(z_0,w_1),\\ldots,(z_0,w_d)$ to $f(z_0,w)=0$, choose $0<\\delta\\leq 1$ and set $z'=\\delta z_1+(1-\\delta)z_0$.\n",
    "\n",
    "* Use Newton iteration starting with $(z,w)=(z',w_i)$ for the equation $f(z,w)=0$ which, _if it converges_ gives a solution $(z',w'_i)$.\n",
    "\n",
    "* For $\\epsilon=\\frac{1}{3}\\min_{k\\neq l}\\left|w_k-w_l\\right|$, there is a bound $\\Delta>0$ such that if $\\delta\\leq \\Delta$, we have $\\left|w'_i-w_i\\right|<\\epsilon$ (hence, we can reliably distinguish the different branches).\n",
    "\n",
    "* For small enough $\\delta$, Newton iteration will converge, so if convergence fails, or escapes the disc $|w-w_i|<\\epsilon$, then just halve $\\delta$.\n",
    "\n",
    "**Explicit bounds:** (S. Kranich, ArXiv:1505.03432) gives an explicit formula for $\\Delta$ in terms of $z_0$ and the coefficients of $f$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "deletable": true,
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   },
   "outputs": [
    {
     "data": {
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IyEhcvHixyesURYHFYvF4n3HjxuHee+9FQkICxowZgy1btqCsrAwbNmwwKnqD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\nAklE5AELJBGRByyQREQe/D+Azwo5WT2ICQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "Graphics object consisting of 2 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "S.plot_paths().show(aspect_ratio=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<iframe srcdoc=\"<!DOCTYPE html>\n",
       "<html>\n",
       "<head>\n",
       "<title></title>\n",
       "<meta charset=&quot;utf-8&quot;>\n",
       "<meta name=viewport content=&quot;width=device-width, user-scalable=no, minimum-scale=1.0, maximum-scale=1.0&quot;>\n",
       "<style>\n",
       "\n",
       "    body { margin: 0px; overflow: hidden; }\n",
       "  \n",
       "</style>\n",
       "</head>\n",
       "\n",
       "<body>\n",
       "\n",
       "<script src=&quot;/nbextensions/threejs/three.min.js&quot;></script>\n",
       "<script src=&quot;/nbextensions/threejs/OrbitControls.js&quot;></script>\n",
       "<script>\n",
       "  if ( !window.THREE ) document.write('\n",
       "<script src=&quot;https://cdn.rawgit.com/mrdoob/three.js/r80/build/three.min.js&quot;><\\/script>\n",
       "<script src=&quot;https://cdn.rawgit.com/mrdoob/three.js/r80/examples/js/controls/OrbitControls.js&quot;><\\/script>\n",
       "            ');\n",
       "</script>\n",
       "        \n",
       "<script>\n",
       "\n",
       "    var scene = new THREE.Scene();\n",
       "\n",
       "    var renderer = new THREE.WebGLRenderer( { antialias: true } );\n",
       "    renderer.setSize( window.innerWidth, window.innerHeight );\n",
       "    renderer.setClearColor( 0xffffff, 1 );\n",
       "    document.body.appendChild( renderer.domElement );\n",
       "\n",
       "    var options = {&quot;aspect_ratio&quot;: [1.0, 1.0, 1.0], &quot;decimals&quot;: 2, &quot;frame&quot;: true, &quot;axes&quot;: false, &quot;axes_labels&quot;: [&quot;x&quot;, &quot;y&quot;, &quot;z&quot;]};\n",
       "\n",
       "    // When animations are supported by the viewer, the value 'false'\n",
       "    // will be replaced with an option set in Python by the user\n",
       "    var animate = false; // options.animate;\n",
       "\n",
       "    var b = [{&quot;x&quot;:-1.25, &quot;y&quot;:-1.44337567297, &quot;z&quot;:-1.99926542966}, {&quot;x&quot;:1.25, &quot;y&quot;:1.44337567297, &quot;z&quot;:1.99926542966}]; // bounds\n",
       "\n",
       "    if ( b[0].x === b[1].x ) {\n",
       "        b[0].x -= 1;\n",
       "        b[1].x += 1;\n",
       "    }\n",
       "    if ( b[0].y === b[1].y ) {\n",
       "        b[0].y -= 1;\n",
       "        b[1].y += 1;\n",
       "    }\n",
       "    if ( b[0].z === b[1].z ) {\n",
       "        b[0].z -= 1;\n",
       "        b[1].z += 1;\n",
       "    }\n",
       "\n",
       "    var rRange = Math.sqrt( Math.pow( b[1].x - b[0].x, 2 )\n",
       "                            + Math.pow( b[1].x - b[0].x, 2 ) );\n",
       "    var xRange = b[1].x - b[0].x;\n",
       "    var yRange = b[1].y - b[0].y;\n",
       "    var zRange = b[1].z - b[0].z;\n",
       "\n",
       "    var ar = options.aspect_ratio;\n",
       "    var a = [ ar[0], ar[1], ar[2] ]; // aspect multipliers\n",
       "    var autoAspect = 2.5;\n",
       "    if ( zRange > autoAspect * rRange && a[2] === 1 ) a[2] = autoAspect * rRange / zRange;\n",
       "\n",
       "    var xMid = ( b[0].x + b[1].x ) / 2;\n",
       "    var yMid = ( b[0].y + b[1].y ) / 2;\n",
       "    var zMid = ( b[0].z + b[1].z ) / 2;\n",
       "\n",
       "    var box = new THREE.Geometry();\n",
       "    box.vertices.push( new THREE.Vector3( a[0]*b[0].x, a[1]*b[0].y, a[2]*b[0].z ) );\n",
       "    box.vertices.push( new THREE.Vector3( a[0]*b[1].x, a[1]*b[1].y, a[2]*b[1].z ) );\n",
       "    var boxMesh = new THREE.LineSegments( box );\n",
       "    if ( options.frame ) scene.add( new THREE.BoxHelper( boxMesh, 'black' ) );\n",
       "\n",
       "    if ( options.axes_labels ) {\n",
       "        var d = options.decimals; // decimals\n",
       "        var offsetRatio = 0.1;\n",
       "        var al = options.axes_labels;\n",
       "\n",
       "        var offset = offsetRatio * a[1]*( b[1].y - b[0].y );\n",
       "        var xm = xMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(xm) ) xm = xm.substr(1);\n",
       "        addLabel( al[0] + '=' + xm, a[0]*xMid, a[1]*b[1].y+offset, a[2]*b[0].z );\n",
       "        addLabel( ( b[0].x ).toFixed(d), a[0]*b[0].x, a[1]*b[1].y+offset, a[2]*b[0].z );\n",
       "        addLabel( ( b[1].x ).toFixed(d), a[0]*b[1].x, a[1]*b[1].y+offset, a[2]*b[0].z );\n",
       "\n",
       "        var offset = offsetRatio * a[0]*( b[1].x - b[0].x );\n",
       "        var ym = yMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(ym) ) ym = ym.substr(1);\n",
       "        addLabel( al[1] + '=' + ym, a[0]*b[1].x+offset, a[1]*yMid, a[2]*b[0].z );\n",
       "        addLabel( ( b[0].y ).toFixed(d), a[0]*b[1].x+offset, a[1]*b[0].y, a[2]*b[0].z );\n",
       "        addLabel( ( b[1].y ).toFixed(d), a[0]*b[1].x+offset, a[1]*b[1].y, a[2]*b[0].z );\n",
       "\n",
       "        var offset = offsetRatio * a[1]*( b[1].y - b[0].y );\n",
       "        var zm = zMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(zm) ) zm = zm.substr(1);\n",
       "        addLabel( al[2] + '=' + zm, a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*zMid );\n",
       "        addLabel( ( b[0].z ).toFixed(d), a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*b[0].z );\n",
       "        addLabel( ( b[1].z ).toFixed(d), a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*b[1].z );\n",
       "    }\n",
       "\n",
       "    function addLabel( text, x, y, z ) {\n",
       "        var fontsize = 14;\n",
       "\n",
       "        var canvas = document.createElement( 'canvas' );\n",
       "        canvas.width = 128;\n",
       "        canvas.height = 32; // powers of two\n",
       "\n",
       "        var context = canvas.getContext( '2d' );\n",
       "        context.fillStyle = 'black';\n",
       "        context.font = fontsize + 'px monospace';\n",
       "        context.textAlign = 'center';\n",
       "        context.textBaseline = 'middle';\n",
       "        context.fillText( text, .5*canvas.width, .5*canvas.height );\n",
       "\n",
       "        var texture = new THREE.Texture( canvas );\n",
       "        texture.needsUpdate = true;\n",
       "\n",
       "        var sprite = new THREE.Sprite( new THREE.SpriteMaterial( { map: texture } ) );\n",
       "        sprite.position.set( x, y, z );\n",
       "        sprite.scale.set( 1, .25 ); // ratio of width to height\n",
       "        scene.add( sprite );\n",
       "    }\n",
       "\n",
       "    if ( options.axes ) scene.add( new THREE.AxisHelper( Math.min( a[0]*b[1].x, a[1]*b[1].y, a[2]*b[1].z ) ) );\n",
       "\n",
       "    var camera = new THREE.PerspectiveCamera( 45, window.innerWidth / window.innerHeight, 0.1, 1000 );\n",
       "    camera.up.set( 0, 0, 1 );\n",
       "    camera.position.set( a[0]*(xMid+xRange), a[1]*(yMid+yRange), a[2]*(zMid+zRange) );\n",
       "\n",
       "    var lights = [{&quot;x&quot;:-5, &quot;y&quot;:3, &quot;z&quot;:0, &quot;color&quot;:&quot;#7f7f7f&quot;, &quot;parent&quot;:&quot;camera&quot;}];\n",
       "    for ( var i=0 ; i < lights.length ; i++ ) {\n",
       "        var light = new THREE.DirectionalLight( lights[i].color, 1 );\n",
       "        light.position.set( a[0]*lights[i].x, a[1]*lights[i].y, a[2]*lights[i].z );\n",
       "        if ( lights[i].parent === 'camera' ) {\n",
       "            light.target.position.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "            scene.add( light.target );\n",
       "            camera.add( light );\n",
       "        } else scene.add( light );\n",
       "    }\n",
       "    scene.add( camera );\n",
       "\n",
       "    var ambient = {&quot;color&quot;:&quot;#7f7f7f&quot;};\n",
       "    scene.add( new THREE.AmbientLight( ambient.color, 1 ) );\n",
       "\n",
       "    var controls = new THREE.OrbitControls( camera, renderer.domElement );\n",
       "    controls.target.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "    controls.addEventListener( 'change', function() { if ( !animate ) render(); } );\n",
       "\n",
       "    window.addEventListener( 'resize', function() {\n",
       "        \n",
       "        renderer.setSize( window.innerWidth, window.innerHeight );\n",
       "        camera.aspect = window.innerWidth / window.innerHeight;\n",
       "        camera.updateProjectionMatrix();\n",
       "        if ( !animate ) render();\n",
       "        \n",
       "    } );\n",
       "\n",
       "    var texts = [];\n",
       "    for ( var i=0 ; i < texts.length ; i++ )\n",
       "        addLabel( texts[i].text, a[0]*texts[i].x, a[1]*texts[i].y, a[2]*texts[i].z );\n",
       "\n",
       "    var points = [{&quot;point&quot;:[1.2499999999999998, 0.7216878364870318, -1.657426396028086], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2499999999999998, 0.7216878364870318, 1.6574263960280866], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.75, 0.14433756729740643, -0.18654372249843593], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.75, 0.14433756729740643, 0.18654372249843598], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.1102230246251565e-16, 1.443375672974064, -0.3418390336361817], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.1102230246251565e-16, 1.443375672974064, 0.34183903363618173], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.2500000000000001, 0.721687836487032, -0.7862136438252054], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.2500000000000001, 0.721687836487032, 0.7862136438252054], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.25, -0.7216878364870318, -1.999265429664268], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.25, -0.7216878364870318, 1.999265429664268], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, -0.5773502691896256, 0.9727573663236411], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, -0.5773502691896256, -0.9727573663236408], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.2499999999999998, 0.7216878364870323, 1.9992654296642682], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.2499999999999998, 0.7216878364870323, -1.999265429664268], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, 0.5773502691896257, -0.9727573663236412], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, 0.5773502691896257, 0.9727573663236411], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-9.992007221626409e-16, -1.443375672974063, 0.3418390336361845], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-9.992007221626409e-16, -1.443375672974063, -0.3418390336361845], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.25, -0.721687836487032, 0.7862136438252052], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.25, -0.721687836487032, -0.7862136438252052], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2500000000000009, -0.7216878364870335, 1.6574263960280913], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2500000000000009, -0.7216878364870335, -1.6574263960280908], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.7499999999999999, -0.14433756729740654, 0.1865437224984361], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.7499999999999999, -0.14433756729740654, -0.1865437224984361], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2499999999999998, 0.7216878364870318, -1.657426396028086], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2499999999999998, 0.7216878364870318, 1.6574263960280866], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.75, 0.14433756729740643, -0.18654372249843593], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.75, 0.14433756729740643, 0.18654372249843598], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.1102230246251565e-16, 1.443375672974064, -0.3418390336361817], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.1102230246251565e-16, 1.443375672974064, 0.34183903363618173], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.2500000000000001, 0.721687836487032, -0.7862136438252054], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.2500000000000001, 0.721687836487032, 0.7862136438252054], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.25, -0.7216878364870318, -1.999265429664268], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.25, -0.7216878364870318, 1.999265429664268], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, -0.5773502691896256, 0.9727573663236411], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, -0.5773502691896256, -0.9727573663236408], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.2499999999999998, 0.7216878364870323, 1.9992654296642682], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.2499999999999998, 0.7216878364870323, -1.999265429664268], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, 0.5773502691896257, -0.9727573663236412], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, 0.5773502691896257, 0.9727573663236411], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-9.992007221626409e-16, -1.443375672974063, 0.3418390336361845], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-9.992007221626409e-16, -1.443375672974063, -0.3418390336361845], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.25, -0.721687836487032, 0.7862136438252052], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.25, -0.721687836487032, -0.7862136438252052], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2500000000000009, -0.7216878364870335, 1.6574263960280913], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2500000000000009, -0.7216878364870335, -1.6574263960280908], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.7499999999999999, -0.14433756729740654, 0.1865437224984361], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.7499999999999999, -0.14433756729740654, -0.1865437224984361], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2499999999999998, 0.7216878364870318, -1.657426396028086], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2499999999999998, 0.7216878364870318, 1.6574263960280866], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.75, 0.14433756729740643, -0.18654372249843593], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.75, 0.14433756729740643, 0.18654372249843598], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.1102230246251565e-16, 1.443375672974064, -0.3418390336361817], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.1102230246251565e-16, 1.443375672974064, 0.34183903363618173], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.2500000000000001, 0.721687836487032, -0.7862136438252054], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.2500000000000001, 0.721687836487032, 0.7862136438252054], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.25, -0.7216878364870318, -1.999265429664268], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.25, -0.7216878364870318, 1.999265429664268], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, -0.5773502691896256, 0.9727573663236411], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, -0.5773502691896256, -0.9727573663236408], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.2499999999999998, 0.7216878364870323, 1.9992654296642682], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.2499999999999998, 0.7216878364870323, -1.999265429664268], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, 0.5773502691896257, -0.9727573663236412], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, 0.5773502691896257, 0.9727573663236411], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-9.992007221626409e-16, -1.443375672974063, 0.3418390336361845], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-9.992007221626409e-16, -1.443375672974063, -0.3418390336361845], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.25, -0.721687836487032, 0.7862136438252052], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.25, -0.721687836487032, -0.7862136438252052], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, 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&quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, 0.5773502691896257, -0.9727573663236412], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, 0.5773502691896257, 0.9727573663236411], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-9.992007221626409e-16, -1.443375672974063, 0.3418390336361845], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-9.992007221626409e-16, -1.443375672974063, -0.3418390336361845], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.25, -0.721687836487032, 0.7862136438252052], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.25, -0.721687836487032, -0.7862136438252052], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2500000000000009, -0.7216878364870335, 1.6574263960280913], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2500000000000009, -0.7216878364870335, -1.6574263960280908], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.7499999999999999, -0.14433756729740654, 0.1865437224984361], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.7499999999999999, -0.14433756729740654, -0.1865437224984361], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2499999999999998, 0.7216878364870318, -1.657426396028086], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2499999999999998, 0.7216878364870318, 1.6574263960280866], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.75, 0.14433756729740643, -0.18654372249843593], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.75, 0.14433756729740643, 0.18654372249843598], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.1102230246251565e-16, 1.443375672974064, -0.3418390336361817], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.1102230246251565e-16, 1.443375672974064, 0.34183903363618173], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.2500000000000001, 0.721687836487032, -0.7862136438252054], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.2500000000000001, 0.721687836487032, 0.7862136438252054], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.25, -0.7216878364870318, -1.999265429664268], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.25, -0.7216878364870318, 1.999265429664268], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, -0.5773502691896256, 0.9727573663236411], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, -0.5773502691896256, -0.9727573663236408], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.2499999999999998, 0.7216878364870323, 1.9992654296642682], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.2499999999999998, 0.7216878364870323, -1.999265429664268], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, 0.5773502691896257, -0.9727573663236412], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, 0.5773502691896257, 0.9727573663236411], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-9.992007221626409e-16, -1.443375672974063, 0.3418390336361845], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-9.992007221626409e-16, -1.443375672974063, -0.3418390336361845], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.25, -0.721687836487032, 0.7862136438252052], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.25, -0.721687836487032, -0.7862136438252052], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2500000000000009, -0.7216878364870335, 1.6574263960280913], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2500000000000009, -0.7216878364870335, -1.6574263960280908], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.7499999999999999, -0.14433756729740654, 0.1865437224984361], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.7499999999999999, -0.14433756729740654, -0.1865437224984361], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2499999999999998, 0.7216878364870318, -1.657426396028086], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2499999999999998, 0.7216878364870318, 1.6574263960280866], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.75, 0.14433756729740643, -0.18654372249843593], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.75, 0.14433756729740643, 0.18654372249843598], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.1102230246251565e-16, 1.443375672974064, -0.3418390336361817], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.1102230246251565e-16, 1.443375672974064, 0.34183903363618173], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.2500000000000001, 0.721687836487032, -0.7862136438252054], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.2500000000000001, 0.721687836487032, 0.7862136438252054], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.25, -0.7216878364870318, -1.999265429664268], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.25, -0.7216878364870318, 1.999265429664268], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, -0.5773502691896256, 0.9727573663236411], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, -0.5773502691896256, -0.9727573663236408], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.2499999999999998, 0.7216878364870323, 1.9992654296642682], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.2499999999999998, 0.7216878364870323, -1.999265429664268], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, 0.5773502691896257, -0.9727573663236412], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, 0.5773502691896257, 0.9727573663236411], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-9.992007221626409e-16, -1.443375672974063, 0.3418390336361845], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-9.992007221626409e-16, -1.443375672974063, -0.3418390336361845], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.25, -0.721687836487032, 0.7862136438252052], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.25, -0.721687836487032, -0.7862136438252052], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2500000000000009, -0.7216878364870335, 1.6574263960280913], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2500000000000009, -0.7216878364870335, -1.6574263960280908], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.7499999999999999, -0.14433756729740654, 0.1865437224984361], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.7499999999999999, -0.14433756729740654, -0.1865437224984361], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2499999999999998, 0.7216878364870318, -1.657426396028086], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2499999999999998, 0.7216878364870318, 1.6574263960280866], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.75, 0.14433756729740643, -0.18654372249843593], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.75, 0.14433756729740643, 0.18654372249843598], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.1102230246251565e-16, 1.443375672974064, -0.3418390336361817], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.1102230246251565e-16, 1.443375672974064, 0.34183903363618173], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.2500000000000001, 0.721687836487032, -0.7862136438252054], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.2500000000000001, 0.721687836487032, 0.7862136438252054], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.25, -0.7216878364870318, -1.999265429664268], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.25, -0.7216878364870318, 1.999265429664268], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, -0.5773502691896256, 0.9727573663236411], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, -0.5773502691896256, -0.9727573663236408], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.2499999999999998, 0.7216878364870323, 1.9992654296642682], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.2499999999999998, 0.7216878364870323, -1.999265429664268], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, 0.5773502691896257, -0.9727573663236412], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.5, 0.5773502691896257, 0.9727573663236411], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-9.992007221626409e-16, -1.443375672974063, 0.3418390336361845], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-9.992007221626409e-16, -1.443375672974063, -0.3418390336361845], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.25, -0.721687836487032, 0.7862136438252052], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.25, -0.721687836487032, -0.7862136438252052], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2500000000000009, -0.7216878364870335, 1.6574263960280913], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.2500000000000009, -0.7216878364870335, -1.6574263960280908], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.7499999999999999, -0.14433756729740654, 0.1865437224984361], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.7499999999999999, -0.14433756729740654, -0.1865437224984361], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1}];\n",
       "    for ( var i=0 ; i < points.length ; i++ ) addPoint( points[i] );\n",
       "\n",
       "    function addPoint( json ) {\n",
       "        var geometry = new THREE.Geometry();\n",
       "        var v = json.point;\n",
       "        geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "\n",
       "        var canvas = document.createElement( 'canvas' );\n",
       "        canvas.width = 128;\n",
       "        canvas.height = 128;\n",
       "\n",
       "        var context = canvas.getContext( '2d' );\n",
       "        context.arc( 64, 64, 64, 0, 2 * Math.PI );\n",
       "        context.fillStyle = json.color;\n",
       "        context.fill();\n",
       "\n",
       "        var texture = new THREE.Texture( canvas );\n",
       "        texture.needsUpdate = true;\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var material = new THREE.PointsMaterial( { size: json.size/100, map: texture,\n",
       "                                                   transparent: transparent, opacity: json.opacity,\n",
       "                                                   alphaTest: .1 } );\n",
       "\n",
       "        var c = geometry.center().multiplyScalar( -1 );\n",
       "        var mesh = new THREE.Points( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        scene.add( mesh );\n",
       "    }\n",
       "\n",
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[0.7335610059926699, 1.0198540286971833, -1.139307170890435], [0.6986704547974039, 1.0399980978219445, -1.0779326153421933], [0.6659912643957882, 1.0588654371972166, -1.0192818980893859], [0.6346730963502222, 1.07694698994885, -0.9627243392416814], [0.6039456033160973, 1.0946875163236243, -0.9074489814569039], [0.5730871546348851, 1.1125036499764964, -0.852475222730058], [0.5413958539132155, 1.1308006309791219, -0.7966355400790609], [0.508162380605217, 1.149987985739601, -0.7385323360812802], [0.4726441489679916, 1.1704944463364924, -0.6764688229414104], [0.43404030136095406, 1.192782388144171, -0.6083519101139118], [0.391467127820866, 1.2173620213477974, -0.5315630304484641], [0.3439336101896437, 1.2448055105477136, -0.44279053217765585], [0.29031687931047523, 1.2757611445538692, -0.33781440454492334], [0.22933735291807844, 1.3109676905315752, -0.2112301362350517], [0.15953303167050964, 1.3512692341944583, -0.05609231891473567], [0.07923165332185242, 1.3976312566003537, 0.13655177231656213], [1.1102230246251565e-16, 1.443375672974064, 0.34183903363618146]], &quot;color&quot;:&quot;blue&quot;, &quot;opacity&quot;:1, &quot;linewidth&quot;:2},{&quot;points&quot;:[[1.2499999999999998, 0.7216878364870318, 1.6574263960280866], [1.1448118261066382, 0.7824182569999292, 1.6155862136628247], [1.0561406277974872, 0.8336125972130843, 1.5558818856224121], [0.9810436306274618, 0.8769698687445312, 1.4870480560627448], [0.9170187455070155, 0.9139346534036557, 1.4146639416695679], [0.8619356224998971, 0.9457369092996206, 1.3421682895944878], [0.8139837857374339, 0.9734219151625657, 1.2715487632895808], [0.7716271653370411, 0.9978765213526954, 1.203812733975328], [0.7335610059926699, 1.0198540286971833, 1.139307170890435], [0.6986704547974039, 1.0399980978219445, 1.0779326153421933], [0.6659912643957882, 1.0588654371972166, 1.0192818980893859], [0.6346730963502222, 1.07694698994885, 0.9627243392416814], [0.6039456033160973, 1.0946875163236243, 0.9074489814569039], [0.5730871546348851, 1.1125036499764964, 0.852475222730058], [0.5413958539132155, 1.1308006309791219, 0.7966355400790609], [0.508162380605217, 1.149987985739601, 0.7385323360812802], [0.4726441489679916, 1.1704944463364924, 0.6764688229414104], [0.43404030136095406, 1.192782388144171, 0.6083519101139118], [0.391467127820866, 1.2173620213477974, 0.5315630304484641], [0.3439336101896437, 1.2448055105477136, 0.44279053217765585], [0.29031687931047523, 1.2757611445538692, 0.33781440454492334], [0.22933735291807844, 1.3109676905315752, 0.2112301362350517], [0.15953303167050964, 1.3512692341944583, 0.05609231891473567], [0.07923165332185242, 1.3976312566003537, -0.13655177231656213], [1.1102230246251565e-16, 1.443375672974064, -0.34183903363618146]], &quot;color&quot;:&quot;blue&quot;, &quot;opacity&quot;:1, &quot;linewidth&quot;:2}];\n",
       "    for ( var i=0 ; i < lines.length ; i++ ) addLine( lines[i] );\n",
       "\n",
       "    function addLine( json ) {\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i < json.points.length - 1 ; i++ ) {\n",
       "            var v = json.points[i];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "            var v = json.points[i+1];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "        }\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var material = new THREE.LineBasicMaterial( { color: json.color, linewidth: json.linewidth,\n",
       "                                                      transparent: transparent, opacity: json.opacity } );\n",
       "\n",
       "        var c = geometry.center().multiplyScalar( -1 );\n",
       "        var mesh = new THREE.LineSegments( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        scene.add( mesh );\n",
       "    }\n",
       "\n",
       "    var surfaces = [];\n",
       "    for ( var i=0 ; i < surfaces.length ; i++ ) addSurface( surfaces[i] );\n",
       "\n",
       "    function addSurface( json ) {\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i < json.vertices.length ; i++ ) {\n",
       "            var v = json.vertices[i];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v.x, a[1]*v.y, a[2]*v.z ) );\n",
       "        }\n",
       "        for ( var i=0 ; i < json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j < f.length - 2 ; j++ ) {\n",
       "                geometry.faces.push( new THREE.Face3( f[0], f[j+1], f[j+2] ) );\n",
       "            }\n",
       "        }\n",
       "        geometry.computeVertexNormals();\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var material = new THREE.MeshPhongMaterial( {\n",
       "                                     color: json.color, side: THREE.DoubleSide,\n",
       "                                     transparent: transparent, opacity: json.opacity,\n",
       "                                     shininess: 20 } );\n",
       "\n",
       "        var c = geometry.center().multiplyScalar( -1 );\n",
       "        var mesh = new THREE.Mesh( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        scene.add( mesh );\n",
       "    }\n",
       "\n",
       "    var scratch = new THREE.Vector3();\n",
       "\n",
       "    function render() {\n",
       "\n",
       "        if ( animate ) requestAnimationFrame( render );\n",
       "        renderer.render( scene, camera );\n",
       "\n",
       "        for ( var i=0 ; i < scene.children.length ; i++ ) {\n",
       "            if ( scene.children[i].type === 'Sprite' ) {\n",
       "                var sprite = scene.children[i];\n",
       "                var adjust = scratch.addVectors( sprite.position, scene.position )\n",
       "                               .sub( camera.position ).length() / 5;\n",
       "                sprite.scale.set( adjust, .25*adjust ); // ratio of canvas width to height\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "    \n",
       "    render();\n",
       "    controls.update();\n",
       "    if ( !animate ) render();\n",
       "\n",
       "</script>\n",
       "\n",
       "</body>\n",
       "</html>\" \n",
       "        width=\"100%\"\n",
       "        height=\"400\"\n",
       "        style=\"border: 0;\">\n",
       "</iframe>\n"
      ],
      "text/plain": [
       "Graphics3d Object"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "S.plot_paths3d(thickness=2).show(viewer=\"threejs\",axes=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<iframe srcdoc=\"<!DOCTYPE html>\n",
       "<html>\n",
       "<head>\n",
       "<title></title>\n",
       "<meta charset=&quot;utf-8&quot;>\n",
       "<meta name=viewport content=&quot;width=device-width, user-scalable=no, minimum-scale=1.0, maximum-scale=1.0&quot;>\n",
       "<style>\n",
       "\n",
       "    body { margin: 0px; overflow: hidden; }\n",
       "  \n",
       "</style>\n",
       "</head>\n",
       "\n",
       "<body>\n",
       "\n",
       "<script src=&quot;/nbextensions/threejs/three.min.js&quot;></script>\n",
       "<script src=&quot;/nbextensions/threejs/OrbitControls.js&quot;></script>\n",
       "<script>\n",
       "  if ( !window.THREE ) document.write('\n",
       "<script src=&quot;https://cdn.rawgit.com/mrdoob/three.js/r80/build/three.min.js&quot;><\\/script>\n",
       "<script src=&quot;https://cdn.rawgit.com/mrdoob/three.js/r80/examples/js/controls/OrbitControls.js&quot;><\\/script>\n",
       "            ');\n",
       "</script>\n",
       "        \n",
       "<script>\n",
       "\n",
       "    var scene = new THREE.Scene();\n",
       "\n",
       "    var renderer = new THREE.WebGLRenderer( { antialias: true } );\n",
       "    renderer.setSize( window.innerWidth, window.innerHeight );\n",
       "    renderer.setClearColor( 0xffffff, 1 );\n",
       "    document.body.appendChild( renderer.domElement );\n",
       "\n",
       "    var options = {&quot;aspect_ratio&quot;: [1.0, 1.0, 1.0], &quot;decimals&quot;: 2, &quot;frame&quot;: true, &quot;axes&quot;: false, &quot;axes_labels&quot;: [&quot;x&quot;, &quot;y&quot;, &quot;z&quot;]};\n",
       "\n",
       "    // When animations are supported by the viewer, the value 'false'\n",
       "    // will be replaced with an option set in Python by the user\n",
       "    var animate = false; // options.animate;\n",
       "\n",
       "    var b = [{&quot;x&quot;:-1.11917226869, &quot;y&quot;:-1.46796384622, &quot;z&quot;:-1.61622815899}, {&quot;x&quot;:1.36511763942, &quot;y&quot;:1.46796384622, &quot;z&quot;:1.61622815899}]; // bounds\n",
       "\n",
       "    if ( b[0].x === b[1].x ) {\n",
       "        b[0].x -= 1;\n",
       "        b[1].x += 1;\n",
       "    }\n",
       "    if ( b[0].y === b[1].y ) {\n",
       "        b[0].y -= 1;\n",
       "        b[1].y += 1;\n",
       "    }\n",
       "    if ( b[0].z === b[1].z ) {\n",
       "        b[0].z -= 1;\n",
       "        b[1].z += 1;\n",
       "    }\n",
       "\n",
       "    var rRange = Math.sqrt( Math.pow( b[1].x - b[0].x, 2 )\n",
       "                            + Math.pow( b[1].x - b[0].x, 2 ) );\n",
       "    var xRange = b[1].x - b[0].x;\n",
       "    var yRange = b[1].y - b[0].y;\n",
       "    var zRange = b[1].z - b[0].z;\n",
       "\n",
       "    var ar = options.aspect_ratio;\n",
       "    var a = [ ar[0], ar[1], ar[2] ]; // aspect multipliers\n",
       "    var autoAspect = 2.5;\n",
       "    if ( zRange > autoAspect * rRange && a[2] === 1 ) a[2] = autoAspect * rRange / zRange;\n",
       "\n",
       "    var xMid = ( b[0].x + b[1].x ) / 2;\n",
       "    var yMid = ( b[0].y + b[1].y ) / 2;\n",
       "    var zMid = ( b[0].z + b[1].z ) / 2;\n",
       "\n",
       "    var box = new THREE.Geometry();\n",
       "    box.vertices.push( new THREE.Vector3( a[0]*b[0].x, a[1]*b[0].y, a[2]*b[0].z ) );\n",
       "    box.vertices.push( new THREE.Vector3( a[0]*b[1].x, a[1]*b[1].y, a[2]*b[1].z ) );\n",
       "    var boxMesh = new THREE.LineSegments( box );\n",
       "    if ( options.frame ) scene.add( new THREE.BoxHelper( boxMesh, 'black' ) );\n",
       "\n",
       "    if ( options.axes_labels ) {\n",
       "        var d = options.decimals; // decimals\n",
       "        var offsetRatio = 0.1;\n",
       "        var al = options.axes_labels;\n",
       "\n",
       "        var offset = offsetRatio * a[1]*( b[1].y - b[0].y );\n",
       "        var xm = xMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(xm) ) xm = xm.substr(1);\n",
       "        addLabel( al[0] + '=' + xm, a[0]*xMid, a[1]*b[1].y+offset, a[2]*b[0].z );\n",
       "        addLabel( ( b[0].x ).toFixed(d), a[0]*b[0].x, a[1]*b[1].y+offset, a[2]*b[0].z );\n",
       "        addLabel( ( b[1].x ).toFixed(d), a[0]*b[1].x, a[1]*b[1].y+offset, a[2]*b[0].z );\n",
       "\n",
       "        var offset = offsetRatio * a[0]*( b[1].x - b[0].x );\n",
       "        var ym = yMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(ym) ) ym = ym.substr(1);\n",
       "        addLabel( al[1] + '=' + ym, a[0]*b[1].x+offset, a[1]*yMid, a[2]*b[0].z );\n",
       "        addLabel( ( b[0].y ).toFixed(d), a[0]*b[1].x+offset, a[1]*b[0].y, a[2]*b[0].z );\n",
       "        addLabel( ( b[1].y ).toFixed(d), a[0]*b[1].x+offset, a[1]*b[1].y, a[2]*b[0].z );\n",
       "\n",
       "        var offset = offsetRatio * a[1]*( b[1].y - b[0].y );\n",
       "        var zm = zMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(zm) ) zm = zm.substr(1);\n",
       "        addLabel( al[2] + '=' + zm, a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*zMid );\n",
       "        addLabel( ( b[0].z ).toFixed(d), a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*b[0].z );\n",
       "        addLabel( ( b[1].z ).toFixed(d), a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*b[1].z );\n",
       "    }\n",
       "\n",
       "    function addLabel( text, x, y, z ) {\n",
       "        var fontsize = 14;\n",
       "\n",
       "        var canvas = document.createElement( 'canvas' );\n",
       "        canvas.width = 128;\n",
       "        canvas.height = 32; // powers of two\n",
       "\n",
       "        var context = canvas.getContext( '2d' );\n",
       "        context.fillStyle = 'black';\n",
       "        context.font = fontsize + 'px monospace';\n",
       "        context.textAlign = 'center';\n",
       "        context.textBaseline = 'middle';\n",
       "        context.fillText( text, .5*canvas.width, .5*canvas.height );\n",
       "\n",
       "        var texture = new THREE.Texture( canvas );\n",
       "        texture.needsUpdate = true;\n",
       "\n",
       "        var sprite = new THREE.Sprite( new THREE.SpriteMaterial( { map: texture } ) );\n",
       "        sprite.position.set( x, y, z );\n",
       "        sprite.scale.set( 1, .25 ); // ratio of width to height\n",
       "        scene.add( sprite );\n",
       "    }\n",
       "\n",
       "    if ( options.axes ) scene.add( new THREE.AxisHelper( Math.min( a[0]*b[1].x, a[1]*b[1].y, a[2]*b[1].z ) ) );\n",
       "\n",
       "    var camera = new THREE.PerspectiveCamera( 45, window.innerWidth / window.innerHeight, 0.1, 1000 );\n",
       "    camera.up.set( 0, 0, 1 );\n",
       "    camera.position.set( a[0]*(xMid+xRange), a[1]*(yMid+yRange), a[2]*(zMid+zRange) );\n",
       "\n",
       "    var lights = [{&quot;x&quot;:-5, &quot;y&quot;:3, &quot;z&quot;:0, &quot;color&quot;:&quot;#7f7f7f&quot;, &quot;parent&quot;:&quot;camera&quot;}];\n",
       "    for ( var i=0 ; i < lights.length ; i++ ) {\n",
       "        var light = new THREE.DirectionalLight( lights[i].color, 1 );\n",
       "        light.position.set( a[0]*lights[i].x, a[1]*lights[i].y, a[2]*lights[i].z );\n",
       "        if ( lights[i].parent === 'camera' ) {\n",
       "            light.target.position.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "            scene.add( light.target );\n",
       "            camera.add( light );\n",
       "        } else scene.add( light );\n",
       "    }\n",
       "    scene.add( camera );\n",
       "\n",
       "    var ambient = {&quot;color&quot;:&quot;#7f7f7f&quot;};\n",
       "    scene.add( new THREE.AmbientLight( ambient.color, 1 ) );\n",
       "\n",
       "    var controls = new THREE.OrbitControls( camera, renderer.domElement );\n",
       "    controls.target.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "    controls.addEventListener( 'change', function() { if ( !animate ) render(); } );\n",
       "\n",
       "    window.addEventListener( 'resize', function() {\n",
       "        \n",
       "        renderer.setSize( window.innerWidth, window.innerHeight );\n",
       "        camera.aspect = window.innerWidth / window.innerHeight;\n",
       "        camera.updateProjectionMatrix();\n",
       "        if ( !animate ) render();\n",
       "        \n",
       "    } );\n",
       "\n",
       "    var texts = [];\n",
       "    for ( var i=0 ; i < texts.length ; i++ )\n",
       "        addLabel( texts[i].text, a[0]*texts[i].x, a[1]*texts[i].y, a[2]*texts[i].z );\n",
       "\n",
       "    var points = [{&quot;point&quot;:[1.3651176394239664, -0.7881510365969342, 0.5830459728553201], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.3651176394239664, -0.7881510365969342, -1.4800955940688787], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.3651176394239664, -0.7881510365969342, 0.897049621213558], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.3651176394239652, 0.7881510365969329, -0.5830459728553197], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.3651176394239652, 0.7881510365969329, 1.480095594068878], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[1.3651176394239652, 0.7881510365969329, -0.8970496212135586], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.7942526201876184, 0.0, 3.573580181622517e-25], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.7942526201876184, 0.0, -1.183749972957155], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.7942526201876184, 0.0, 1.1837499729571548], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.42246258506147544, 5.551115123125783e-17, -2.948692915026061e-17], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.42246258506147544, 5.551115123125783e-17, -1.5399080994210927], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.42246258506147544, 5.551115123125783e-17, 1.5399080994210927], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.1191722686891996, -0.646154410597273, -0.23926841642138502], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.1191722686891996, -0.646154410597273, 0.9662607452094888], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.1191722686891996, -0.646154410597273, -0.7269923287881044], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.1191722686892005, 0.6461544105972742, 0.23926841642138585], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.1191722686892005, 0.6461544105972742, -0.9662607452094899], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.1191722686892005, 0.6461544105972742, 0.7269923287881037], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.26340111345270784, 0.33029454916588913, -1.3710910927872846], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.26340111345270784, 0.33029454916588913, -0.032636328443807], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.26340111345270784, 0.33029454916588913, 1.4037274212310913], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.509456950058773, 0.6388388091203098, -0.9597673374365547], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.509456950058773, 0.6388388091203098, 1.1610675641182038], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.509456950058773, 0.6388388091203098, -0.20130022668164857], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.09400676898947882, 0.4118705660289203, -0.9895213116792239], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.09400676898947882, 0.4118705660289203, 0.058808631781016335], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.09400676898947882, 0.4118705660289203, 0.9307126798982077], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[5.551115123125783e-17, 1.4679638462220748, -1.2069162835185676], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[5.551115123125783e-17, 1.4679638462220748, 1.030030989826201], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[5.551115123125783e-17, 1.4679638462220748, 0.17688529369236733], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.1988606050497552, 0.8712652380581601, -0.9433867063759931], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.1988606050497552, 0.8712652380581601, 0.4364588060602512], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.1988606050497552, 0.8712652380581601, 0.5069279003157414], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.9515640924849185, -0.4582491151093564, -0.1028647018745098], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.9515640924849185, -0.4582491151093564, 0.725045149075389], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.9515640924849185, -0.4582491151093564, -0.6221804472008797], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.3806256369939671, -0.18329964604374238, 0.3059952063133455], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.3806256369939671, -0.18329964604374238, 0.07333316435345837], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.3806256369939671, -0.18329964604374238, -0.37932837066680364], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.9515640924849179, 0.4582491151093564, 0.10286470187450969], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.9515640924849179, 0.4582491151093564, -0.7250451490753888], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.9515640924849179, 0.4582491151093564, 0.6221804472008788], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.38062563699396723, 0.18329964604374271, -0.30599520631334565], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.38062563699396723, 0.18329964604374271, -0.07333316435345855], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.38062563699396723, 0.18329964604374271, 0.3793283706668035], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.1657341758564144e-15, -1.4679638462220752, 1.2069162835185685], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.1657341758564144e-15, -1.4679638462220752, -1.0300309898262012], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-1.1657341758564144e-15, -1.4679638462220752, -0.17688529369236772], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.19886060504975597, -0.87126523805816, 0.9433867063759926], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.19886060504975597, -0.87126523805816, -0.43645880606025084], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.19886060504975597, -0.87126523805816, -0.5069279003157414], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.09400676898947893, -0.4118705660289202, 0.9895213116792234], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.09400676898947893, -0.4118705660289202, -0.0588086317810163], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.09400676898947893, -0.4118705660289202, -0.9307126798982067], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.5094569500587725, -0.6388388091203099, 0.9597673374365545], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.5094569500587725, -0.6388388091203099, -1.1610675641182024], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.5094569500587725, -0.6388388091203099, 0.20130022668164793], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.2634011134527078, -0.3302945491658893, 1.3710910927872852], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.2634011134527078, -0.3302945491658893, 0.03263632844380699], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.2634011134527078, -0.3302945491658893, -1.4037274212310924], &quot;size&quot;:20, 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&quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.19886060504975597, -0.87126523805816, 0.9433867063759926], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.19886060504975597, -0.87126523805816, -0.43645880606025084], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.19886060504975597, -0.87126523805816, -0.5069279003157414], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.09400676898947893, -0.4118705660289202, 0.9895213116792234], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.09400676898947893, -0.4118705660289202, -0.0588086317810163], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[-0.09400676898947893, -0.4118705660289202, -0.9307126798982067], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.5094569500587725, -0.6388388091203099, 0.9597673374365545], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.5094569500587725, -0.6388388091203099, -1.1610675641182024], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.5094569500587725, -0.6388388091203099, 0.20130022668164793], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.2634011134527078, -0.3302945491658893, 1.3710910927872852], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.2634011134527078, -0.3302945491658893, 0.03263632844380699], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1},{&quot;point&quot;:[0.2634011134527078, -0.3302945491658893, -1.4037274212310924], &quot;size&quot;:20, &quot;color&quot;:&quot;purple&quot;, &quot;opacity&quot;:1}];\n",
       "    for ( var i=0 ; i < points.length ; i++ ) addPoint( points[i] );\n",
       "\n",
       "    function addPoint( json ) {\n",
       "        var geometry = new THREE.Geometry();\n",
       "        var v = json.point;\n",
       "        geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "\n",
       "        var canvas = document.createElement( 'canvas' );\n",
       "        canvas.width = 128;\n",
       "        canvas.height = 128;\n",
       "\n",
       "        var context = canvas.getContext( '2d' );\n",
       "        context.arc( 64, 64, 64, 0, 2 * Math.PI );\n",
       "        context.fillStyle = json.color;\n",
       "        context.fill();\n",
       "\n",
       "        var texture = new THREE.Texture( canvas );\n",
       "        texture.needsUpdate = true;\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var material = new THREE.PointsMaterial( { size: json.size/100, map: texture,\n",
       "                                                   transparent: transparent, opacity: json.opacity,\n",
       "                                                   alphaTest: .1 } );\n",
       "\n",
       "        var c = geometry.center().multiplyScalar( -1 );\n",
       "        var mesh = new THREE.Points( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        scene.add( mesh );\n",
       "    }\n",
       "\n",
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[-0.40899172124773236, 0.19696003225531186, 0.3761774186870507], [-0.38062563699396723, 0.18329964604374271, 0.37932837066680397]], &quot;color&quot;:&quot;blue&quot;, &quot;opacity&quot;:1, &quot;linewidth&quot;:2},{&quot;points&quot;:[[1.3651176394239652, 0.7881510365969329, -0.5830459728553197], [1.199013721296022, 0.7591660121397759, -0.6151180763542072], [1.0681607887849482, 0.7363322614861474, -0.6501334792571117], [0.9625609113500879, 0.717905152332921, -0.6870752028950508], [0.8749703671057769, 0.7026206599756638, -0.7252734736512345], [0.800038555153787, 0.689545108077035, -0.7643287910764166], [0.7337160736599224, 0.6779718786804936, -0.8040289059444873], [0.6728386041869495, 0.6673488003000981, -0.8442719998900103], [0.6148265794422145, 0.6572257401400111, -0.8849896178155026], [0.557461563789625, 0.6472155826409282, -0.9260509962704336], [0.509456950058773, 0.6388388091203098, -0.9597673374365547]], &quot;color&quot;:&quot;blue&quot;, &quot;opacity&quot;:1, 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-1.3254152398702357, -0.2385588471739304], [-0.08590602532138394, -1.210195316706951, -0.29635025045028035], [-0.11788113173914885, -1.1142512181229218, -0.3506432296351074], [-0.14541224035987446, -1.0316417229386627, -0.40175597054267254], [-0.17000028928386104, -0.9578631353358633, -0.44990641573009027], [-0.19282414589446426, -0.889378160802711, -0.4951435142881015], [-0.19886060504975597, -0.87126523805816, -0.5069279003157419]], &quot;color&quot;:&quot;blue&quot;, &quot;opacity&quot;:1, &quot;linewidth&quot;:2},{&quot;points&quot;:[[1.3651176394239664, -0.7881510365969342, 0.5830459728553201], [1.2662086007557407, -0.6515946550911104, 0.525992325034305], [1.1888088975239266, -0.544734619392911, 0.47709734491661027], [1.1270923227984944, -0.4595271172584652, 0.43328625573343765], [1.0768241272235697, -0.39012554400524146, 0.39243354546806786], [1.0347841869233734, -0.33208411317734365, 0.35300482162648517], [0.9985707404234625, -0.28208689065382536, 0.3139170495076514], [0.9663491271711613, -0.23760089653984273, 0.27436011820254347], [0.9366728318292206, -0.1966290342817636, 0.23366949269325493], [0.9083528460997622, -0.15752972821648523, 0.19123736332644828], [0.8803590489490545, -0.11888076655453406, 0.1464499649337238], [0.8517426980082748, -0.07937229099972111, 0.09864001362931803], [0.8215730644001326, -0.03771931314899297, 0.047044729690513445], [0.7942526201876184, 0.0, 0.0]], &quot;color&quot;:&quot;blue&quot;, &quot;opacity&quot;:1, &quot;linewidth&quot;:2},{&quot;points&quot;:[[1.3651176394239664, -0.7881510365969342, -1.4800955940688787], [1.2662086007557407, -0.6515946550911104, -1.4160453299766644], [1.1888088975239266, -0.544734619392911, -1.3697181968111911], [1.1270923227984944, -0.4595271172584652, -1.3356764399697636], [1.0768241272235697, -0.39012554400524146, -1.3100219529495332], [1.0347841869233734, -0.33208411317734365, -1.28999296551169], [0.9985707404234625, -0.28208689065382536, -1.2736947808266088], [0.9663491271711613, 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-0.23760089653984273, 0.9854474013230431], [0.9366728318292206, -0.1966290342817636, 1.0137028570245679], [0.9083528460997622, -0.15752972821648523, 1.044398791628085], [0.8803590489490545, -0.11888076655453406, 1.0774791380937039], [0.8517426980082748, -0.07937229099972111, 1.1129078015403426], [0.8215730644001326, -0.03771931314899297, 1.150566706628727], [0.7942526201876184, 0.0, 1.1837499729571554]], &quot;color&quot;:&quot;blue&quot;, &quot;opacity&quot;:1, &quot;linewidth&quot;:2}];\n",
       "    for ( var i=0 ; i < lines.length ; i++ ) addLine( lines[i] );\n",
       "\n",
       "    function addLine( json ) {\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i < json.points.length - 1 ; i++ ) {\n",
       "            var v = json.points[i];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "            var v = json.points[i+1];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "        }\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var material = new THREE.LineBasicMaterial( { color: json.color, linewidth: json.linewidth,\n",
       "                                                      transparent: transparent, opacity: json.opacity } );\n",
       "\n",
       "        var c = geometry.center().multiplyScalar( -1 );\n",
       "        var mesh = new THREE.LineSegments( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        scene.add( mesh );\n",
       "    }\n",
       "\n",
       "    var surfaces = [];\n",
       "    for ( var i=0 ; i < surfaces.length ; i++ ) addSurface( surfaces[i] );\n",
       "\n",
       "    function addSurface( json ) {\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i < json.vertices.length ; i++ ) {\n",
       "            var v = json.vertices[i];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v.x, a[1]*v.y, a[2]*v.z ) );\n",
       "        }\n",
       "        for ( var i=0 ; i < json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j < f.length - 2 ; j++ ) {\n",
       "                geometry.faces.push( new THREE.Face3( f[0], f[j+1], f[j+2] ) );\n",
       "            }\n",
       "        }\n",
       "        geometry.computeVertexNormals();\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var material = new THREE.MeshPhongMaterial( {\n",
       "                                     color: json.color, side: THREE.DoubleSide,\n",
       "                                     transparent: transparent, opacity: json.opacity,\n",
       "                                     shininess: 20 } );\n",
       "\n",
       "        var c = geometry.center().multiplyScalar( -1 );\n",
       "        var mesh = new THREE.Mesh( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        scene.add( mesh );\n",
       "    }\n",
       "\n",
       "    var scratch = new THREE.Vector3();\n",
       "\n",
       "    function render() {\n",
       "\n",
       "        if ( animate ) requestAnimationFrame( render );\n",
       "        renderer.render( scene, camera );\n",
       "\n",
       "        for ( var i=0 ; i < scene.children.length ; i++ ) {\n",
       "            if ( scene.children[i].type === 'Sprite' ) {\n",
       "                var sprite = scene.children[i];\n",
       "                var adjust = scratch.addVectors( sprite.position, scene.position )\n",
       "                               .sub( camera.position ).length() / 5;\n",
       "                sprite.scale.set( adjust, .25*adjust ); // ratio of canvas width to height\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "    \n",
       "    render();\n",
       "    controls.update();\n",
       "    if ( !animate ) render();\n",
       "\n",
       "</script>\n",
       "\n",
       "</body>\n",
       "</html>\" \n",
       "        width=\"100%\"\n",
       "        height=\"400\"\n",
       "        style=\"border: 0;\">\n",
       "</iframe>\n"
      ],
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       "Graphics3d Object"
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   ],
   "source": [
    "f=z*w^3+z^3+w\n",
    "S=RiemannSurface(f)\n",
    "S.plot_paths3d(thickness=2).show(viewer=\"threejs\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Monodromy generators\n",
    "\n",
    "This lifted graph contains generators of the monodromy. In particular, we can compute the local monodromy by composing the computed permutations along the loops around the branch locus"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
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    {
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       "<div class=\"notruncate\">\n",
       "<table  class=\"table_form\">\n",
       "<tbody>\n",
       "<tr>\n",
       "<th>Branch point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .</th>\n",
       "<th>local monodromy</th>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">0.0</script></td>\n",
       "<td><script type=\"math/tex\">(0,2)</script></td>\n",
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       "<td><script type=\"math/tex\">-0.761251273988 - 2.77555756156 \\times 10^{-17}i</script></td>\n",
       "<td><script type=\"math/tex\">(1,2)</script></td>\n",
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       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">-0.474632405983 - 0.595170212073i</script></td>\n",
       "<td><script type=\"math/tex\">(0,1)</script></td>\n",
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       "<tr class =\"row-b\">\n",
       "<td><script type=\"math/tex\">-0.474632405983 + 0.595170212073i</script></td>\n",
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       "<td><script type=\"math/tex\">0.169394344463 - 0.742165115195i</script></td>\n",
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       "<td><script type=\"math/tex\">0.685863698514 - 0.330294549166i</script></td>\n",
       "<td><script type=\"math/tex\">(0,1)</script></td>\n",
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       "<td><script type=\"math/tex\">0.685863698514 + 0.330294549166i</script></td>\n",
       "<td><script type=\"math/tex\">(1,2)</script></td>\n",
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       "<tr class =\"row-a\">\n",
       "<td><script type=\"math/tex\">\\infty</script></td>\n",
       "<td><script type=\"math/tex\">(0,2,1)</script></td>\n",
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      "text/plain": [
       "  Branch point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   local monodromy\n",
       "+------------------------------------------------------------------------------------+-----------------+\n",
       "  0.0                                                                                  (0,2)\n",
       "  -0.761251273988 - 2.77555756156e-17*I                                                (1,2)\n",
       "  -0.474632405983 - 0.595170212073*I                                                   (0,1)\n",
       "  -0.474632405983 + 0.595170212073*I                                                   (0,1)\n",
       "  0.169394344463 - 0.742165115195*I                                                    (1,2)\n",
       "  0.169394344463 + 0.742165115195*I                                                    (0,1)\n",
       "  0.685863698514 - 0.330294549166*I                                                    (0,1)\n",
       "  0.685863698514 + 0.330294549166*I                                                    (1,2)\n",
       "  Infinity                                                                             (0,2,1)"
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   "source": [
    "table([[\"Branch point\"+\" .\"*35,\"local monodromy\"]]+zip(S.branch_locus + [unsigned_infinity], S.monodromy_group()),header_row=True)"
   ]
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   "cell_type": "markdown",
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   "source": [
    "### Homology basis\n",
    "\n",
    " * A cycle basis for the lifted graph generates $H_1(C,\\mathbb{Z})$.\n",
    " * $C(\\mathbb{C})$ is an _oriented_ $2$-dimensional manifold, so $1$-cycles have a _signed_ intersection pairing:\n",
    " $\\alpha,\\beta$\n",
    " <table>\n",
    " <tr>\n",
    " <td><img src=\"intersection1.svg\" style=\"width:100px\"></img></td>\n",
    " <td><img src=\"intersection2.svg\" style=\"width:100px\"></img></td>\n",
    " <td><img src=\"intersection3.svg\" style=\"width:100px\"></img></td>\n",
    " </tr>\n",
    " <tr>\n",
    " <td>$\\langle \\alpha,\\beta\\rangle_P=1$</td>\n",
    " <td>$\\langle \\alpha,\\beta\\rangle_P=-1$</td>\n",
    " <td>$\\langle \\alpha,\\beta\\rangle_P=-\\frac{1}{2}$</td>\n",
    " </tr>\n",
    " </table>\n",
    " \n",
    " $$\\langle \\alpha,\\beta \\rangle=\\sum_{P\\in \\alpha\\cap\\beta} \\langle \\alpha,\\beta \\rangle_P$$\n",
    " * This defines an _alternating_ pairing on $H_1(C,\\mathbb{Z})$. We can compute its Gram matrix on the cycle basis.\n",
    " \n",
    "**Frobenius algorithm:** Brings an alternating pairing in standard form:\n",
    "$$\\begin{pmatrix}\n",
    "0&c_1\\\\\n",
    "-c_1&0\\\\\n",
    "&&\\ddots\\\\\n",
    "&&&0&c_g\\\\\n",
    "&&&-c_g&0\\\\\n",
    "&&&&&0\\\\\n",
    "&&&&&&\\ddots\\\\\n",
    "&&&&&&&0\n",
    "\\end{pmatrix}$$\n",
    " \n",
    "**Theorem:** This pairing is _nondegenerate_ on $H_1(C,\\mathbb{Z})$, with $c_1=\\cdots=c_g=1$. Hence, there is a basis $\\alpha_1\\ldots,\\alpha_g,\\beta_1,\\ldots,\\beta_g$ with\n",
    "$\\langle \\alpha_i,\\beta_j\\rangle=\\delta_{ij}$ and $\\langle \\alpha_i,\\alpha_j\\rangle=\\langle \\beta_i,\\beta_j\\rangle=0$.\n",
    "\n",
    "Using the Frobenius algorithm, we find such a basis as linear combination of a cycle basis of the upstairs graph."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "source": [
    "## 3. Finding a cohomology basis\n",
    "Holomorphic differentials on $C$ can be expressed in the form\n",
    "$$h(z,w)\\frac{df\\, dz}{dw}\\text{ where }\\mathrm{totdeg}(h)\\leq \\mathrm{totdeg}(f)-3$$\n",
    "If $C$ is nonsingular, then any such differential is holomorphic. For singularities one needs to impose conditions on $h(z,w)$ so that the pullback along the relevant blow-ups is regular. This is implemented in Singular for curves over $\\mathbb{Q}$.\n",
    "\n",
    "**TODO:** Provide a general implementation in Sage.\n",
    "\n",
    "(Presently, the code works for curves defined over any exact field embedded in $\\mathbb{C}$, but for field different from $\\mathbb{Q}$ it needs a basis for the relevant space of polynomials $h$)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, w, z]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S.cohomology_basis()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## 4. Compute the integrals\n",
    "\n",
    "We only have to integrate differentials $h(z,w)\\frac{df\\, dz}{dw}$ along straight lines $z=(1-t)z_0+tz_1$, and from the analytic continuation we already have a lot of sample points for $w$, which we can use to interpolate initial values for Newton iteration to compute $w(t)$.\n",
    "\n",
    "Since we stay away from branch points and singularities, we know the integrand behaves nicely. Gauss-Legendre is probably a good integration scheme. We repeatedly double the number of evaluation points and estimate the error based on that."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrr}\n",
       "1.38777878078 \\times 10^{-16} + 2.7781638803i & -0.775197805904 - 0.618199621328i & 0.430202326363 - 0.893324335547i & -1.74185365871 + 1.38908194015i & 1.73472347598 \\times 10^{-16} + 0.550249428438i & 0.430202326363 + 0.343074907109i \\\\\n",
       "1.82145964978 \\times 10^{-16} + 1.78664867109i & 0.536453526446 + 1.11395722593i & -1.20540013227 - 0.275124714219i & -0.430202326363 + 0.893324335547i & 2.49800180541 \\times 10^{-16} + 2.7781638803i & -1.20540013227 - 2.50303916608i \\\\\n",
       "1.11022302463 \\times 10^{-16} + 0.550249428438i & 2.17205598507 - 0.495757604603i & 1.74185365871 - 1.38908194015i & 1.20540013227 + 0.275124714219i & 1.38777878078 \\times 10^{-16} + 1.78664867109i & 1.74185365871 - 0.397566730944i\n",
       "\\end{array}\\right)</script></html>"
      ],
      "text/plain": [
       "[1.3877787807814457e-16 + 2.7781638803006663*I    -0.7751978059036679 - 0.6181996213282022*I      0.430202326362528 - 0.8933243355469873*I     -1.741853658712075 + 1.3890819401503331*I  1.734723475976807e-16 + 0.5502494284375682*I    0.43020232636252753 + 0.3430749071094185*I]\n",
       "[1.8214596497756474e-16 + 1.7866486710939695*I     0.5364535264458775 + 1.1139572259315484*I   -1.2054001322661958 - 0.27512471421878293*I    -0.4302023263625278 + 0.8933243355469844*I  2.498001805406602e-16 + 2.7781638803006627*I    -1.2054001322661956 - 2.5030391660818814*I]\n",
       "[1.1102230246251565e-16 + 0.5502494284375674*I     2.1720559850746017 - 0.4957576046033465*I      1.741853658712073 - 1.3890819401503318*I    1.2054001322661962 + 0.27512471421878365*I 1.3877787807814457e-16 + 1.7866486710939715*I      1.7418536587120743 - 0.397566730943639*I]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show(S.period_matrix())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Riemann matrix\n",
    "\n",
    "The period matrix, given a basis of holomorphic differentials $\\omega_1,\\ldots,\\omega_g$ and a homology basis $\\alpha_1,\\ldots,\\alpha_g,\\beta_1,\\ldots,\\beta_g$, has the shape:\n",
    "$$\\left(\\begin{array}{ccc|ccc}\n",
    "\\int_{\\alpha_1} \\omega_1&\\cdots&\\int_{\\alpha_g} \\omega_1&\\int_{\\beta_1} \\omega_1&\\cdots&\\int_{\\beta_g} \\omega_1\\\\\n",
    "\\vdots&\\ddots&\\vdots&\\vdots&\\ddots&\\vdots\\\\\n",
    "\\int_{\\alpha_1} \\omega_g&\\cdots&\\int_{\\alpha_g} \\omega_g&\\int_{\\beta_1} \\omega_g&\\cdots&\\int_{\\beta_g} \\omega_g\\\\\n",
    "\\end{array}\\right)=\\left( M_\\alpha \\mid M_\\beta \\right)=M_\\alpha\\left(I_g \\mid M_\\alpha^{-1}M_\\beta\\right)$$\n",
    "\n",
    "So we can use a change of basis of our differentials to get a square matrix $M=M_\\alpha^{-1}M_\\beta$ which only depends on our choice of \n",
    "\n",
    "**Theorem** (Riemann): This matrix is symmetric and its imaginary part is positive definite."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n",
       "0.5 + 0.566946709514i & 0.5 - 0.188982236505i & 2.77555756156 \\times 10^{-16} + 0.377964473009i \\\\\n",
       "0.5 - 0.188982236505i & 0.5 + 0.944911182523i & -0.5 - 0.566946709514i \\\\\n",
       "7.29738914413 \\times 10^{-17} + 0.377964473009i & -0.5 - 0.566946709514i & 1.0 + 1.13389341903i\n",
       "\\end{array}\\right)</script></html>"
      ],
      "text/plain": [
       "[                  0.5 + 0.5669467095138411*I                  0.5 - 0.18898223650461385*I 2.7755575615628914e-16 + 0.377964473009228*I]\n",
       "[  0.5000000000000003 - 0.18898223650461385*I   0.49999999999999983 + 0.9449111825230684*I   -0.49999999999999944 - 0.566946709513842*I]\n",
       "[7.297389144134398e-17 + 0.3779644730092276*I                  -0.5 - 0.5669467095138409*I    0.9999999999999999 + 1.1338934190276828*I]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show(S.riemann_matrix())\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[7x^{2} - 7x + 4, 7x^{2} - 7x + 2, 7x^{2} + 1, 7x^{2} - 7x + 2, 7x^{2} - 7x + 8, 7x^{2} + 7x + 4, 7x^{2} + 1, 7x^{2} + 7x + 4, 7x^{2} - 14x + 16\\right]</script></html>"
      ],
      "text/plain": [
       "[7*x^2 - 7*x + 4,\n",
       " 7*x^2 - 7*x + 2,\n",
       " 7*x^2 + 1,\n",
       " 7*x^2 - 7*x + 2,\n",
       " 7*x^2 - 7*x + 8,\n",
       " 7*x^2 + 7*x + 4,\n",
       " 7*x^2 + 1,\n",
       " 7*x^2 + 7*x + 4,\n",
       " 7*x^2 - 14*x + 16]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show([c.algdep(4) for c in S.riemann_matrix().list()])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## 5. Applications\n",
    "\n",
    "### Endomorphisms and isogenies\n",
    "In order to find endomorphism it is easiest to consider integer linear transformations on the homology:\n",
    "\n",
    "From:\n",
    "$$\\begin{pmatrix}I\\mid M\\end{pmatrix}\n",
    "\\begin{pmatrix} D&B\\\\C&A\\end{pmatrix}=\n",
    "\\begin{pmatrix} MA+B \\mid MC+D\\end{pmatrix}$$\n",
    "\n",
    "Bringing this in standard form:\n",
    "$$\\begin{pmatrix}\n",
    "I \\mid (MA+B)^{-1}(MC+D)\\end{pmatrix}$$\n",
    "\n",
    "For an endomorphism we would be able to find a transformation $\\begin{pmatrix}D&B\\\\C&A\\end{pmatrix}$ that satisfies the equation\n",
    "$$(MC+D)-(MA+B)M=0$$\n",
    "\n",
    "This gives $g^2$ equation with coefficients in $\\mathbb{C}$ in $4g^2$ integer variables. We can use LLL to find approximate solutions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rr}\n",
       "1 & 0 \\\\\n",
       "0 & 1\n",
       "\\end{array}\\right), \\left(\\begin{array}{rr}\n",
       "0 & -1 \\\\\n",
       "1 & 1\n",
       "\\end{array}\\right)\\right]</script></html>"
      ],
      "text/plain": [
       "[\n",
       "[1 0]  [ 0 -1]\n",
       "[0 1], [ 1  1]\n",
       "]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "S=RiemannSurface(w^2-z^3+1)\n",
    "show(S.endomorphism_basis())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### An example from Drew Sutherland\n",
    "\n",
    "Two genus 3 curves (one hyperelliptic, the other non-hyperelliptic) with isogenous Jacobians (based on L-series computations):\n",
    "$$C_1: -x^2y^2 - xy^3 + x^3 + 2x^2y + 2xy^2 - x^2 - y = 0 $$\n",
    "$$C_2: y^2 + (x^4+x^3+x^2+1)  y = x^7 - 8x^5 - 4x^4 + 18x^3 - 3x^2 - 16x + 8 $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 14.9 s, sys: 18.6 ms, total: 14.9 s\n",
      "Wall time: 15 s\n"
     ]
    }
   ],
   "source": [
    "RR.<x,y>=QQ[]\n",
    "f1=-x^2*y^2 - x*y^3 + x^3 + 2*x^2*y + 2*x*y^2 - x^2 - y\n",
    "f2=y^2 + (x^4+x^3+x^2+1) * y -( x^7 - 8*x^5 - 4*x^4 + 18 * x^3 - 3*x^2 - 16*x + 8)\n",
    "S1=RiemannSurface(f1)\n",
    "S2=RiemannSurface(f2)\n",
    "%time E12 = S1.isogeny_basis(S2); E12"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrrrr}\n",
       "0 & 1 & 0 & 2 & -1 & 1 \\\\\n",
       "-2 & -1 & 2 & 0 & -1 & -1 \\\\\n",
       "0 & 1 & 0 & 0 & -1 & -1 \\\\\n",
       "-2 & -1 & 0 & -2 & 1 & -1 \\\\\n",
       "0 & 1 & 0 & 0 & -1 & 1 \\\\\n",
       "0 & 2 & 2 & 0 & 0 & -2\n",
       "\\end{array}\\right)\\right]</script></html>"
      ],
      "text/plain": [
       "[\n",
       "[ 0  1  0  2 -1  1]\n",
       "[-2 -1  2  0 -1 -1]\n",
       "[ 0  1  0  0 -1 -1]\n",
       "[-2 -1  0 -2  1 -1]\n",
       "[ 0  1  0  0 -1  1]\n",
       "[ 0  2  2  0  0 -2]\n",
       "]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "[1 0 0 0 0 0]\n",
       "[0 2 0 0 0 0]\n",
       "[0 0 2 0 0 0]\n",
       "[0 0 0 2 0 0]\n",
       "[0 0 0 0 2 0]\n",
       "[0 0 0 0 0 4]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "show(E12)\n",
    "E12[0].smith_form()[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-4  0  0  0  0  0]\n",
       "[ 0 -4  0  0  0  0]\n",
       "[ 0  0 -4  0  0  0]\n",
       "[ 0  0  0 -4  0  0]\n",
       "[ 0  0  0  0 -4  0]\n",
       "[ 0  0  0  0  0 -4]"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E21 = S2.isogeny_basis(S1)\n",
    "E12[0]*E21[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Recognizing isogeny factors\n",
    "Kani-Rosen show that idempotents in the endomorphism ring are in bijection to decompositions of an abelian variety. If you can compute endomorphism rings, you can solve for idempotents. The following takes a little too long to execute during the talk:"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "f=4*x^6 - 54*x^5*y - 729*x^4 + 108*x^3*y^3 + 39366*x^2 - 54*x*y^5 - 531441\n",
    "S=RiemannSurface(f,prec=60)\n",
    "%time E=S.endomorphism_basis(); E\n",
    "----\n",
    "CPU times: user 2min 1s, sys: 252 ms, total: 2min 2s\n",
    "Wall time: 2min 2s\n",
    "\n",
    "[\n",
    "[1 0 0 0 0 0 0 0 0 0 0 0]  [-1  0 -1 -1 -1 -1  0 -1  0 -1  0  0]\n",
    "[0 1 0 0 0 0 0 0 0 0 0 0]  [ 0  0 -1  0  0  0  1  0  0 -1  0  0]\n",
    "[0 0 1 0 0 0 0 0 0 0 0 0]  [-1  1  0 -1  0 -1  0  0  0 -1  1  0]\n",
    "[0 0 0 1 0 0 0 0 0 0 0 0]  [-2  3  2 -1  0 -1  1  1  1  0  1 -1]\n",
    "[0 0 0 0 1 0 0 0 0 0 0 0]  [ 1 -1 -1  0  0  1  0  0 -1 -1  0  1]\n",
    "[0 0 0 0 0 1 0 0 0 0 0 0]  [ 1 -1 -1  1  0  0  0  0  0  1 -1  0]\n",
    "[0 0 0 0 0 0 1 0 0 0 0 0]  [ 0  0  0 -1  1  0 -1  0 -1 -2  1  1]\n",
    "[0 0 0 0 0 0 0 1 0 0 0 0]  [ 0  0  1  2  0  1  0  0  1  3 -1 -1]\n",
    "[0 0 0 0 0 0 0 0 1 0 0 0]  [ 0 -1  0  0 -2 -1 -1 -1  0  2 -1 -1]\n",
    "[0 0 0 0 0 0 0 0 0 1 0 0]  [ 1 -2  0  0  1  1 -1  0 -1 -1  0  1]\n",
    "[0 0 0 0 0 0 0 0 0 0 1 0]  [-1  0  2 -1  0 -1 -1  0  0  0  0  0]\n",
    "[0 0 0 0 0 0 0 0 0 0 0 1], [ 0 -1  1 -1  1  0 -1  0 -1 -1  1  0],\n",
    "\n",
    "[-2  2  2  0  0  0  0  0  1  2  0 -1]\n",
    "[-1  2  2  0 -1  0  0  0  1  3  0 -1]\n",
    "[-1 -1  0  0  0 -1 -1 -1  0  1 -1 -1]\n",
    "[-1 -3 -3  0 -1 -1 -2 -3 -1  0 -1  0]\n",
    "[-1  1  2  0  0  0  0  0  1  1  0 -1]\n",
    "[ 0  2  1  0  1  0  1  1  1  0  1  0]\n",
    "[ 0 -3 -1  0  0 -1 -2 -1 -1 -1 -1  0]\n",
    "[ 3  0 -1 -1  0  1  2  2 -1 -3  1  2]\n",
    "[ 1  1  0  0  3  2  2  2  0 -3  2  1]\n",
    "[ 0  1  0  0 -1  0  0  0  0  0  0  0]\n",
    "[ 0  0 -3  1  0  1  0 -1  0 -1  0  1]\n",
    "[ 1 -1 -2  0 -1  0  0  0 -1 -1  0  0]\n",
    "]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We reconstruct the matrices and we solve for the idempotent, i.e., we solve the equation\n",
    "$$(aE_1+bE_2+cE_3)^2=(aE_1+bE_2+cE_3)$$\n",
    "and we compute the characteristic polynomials of the idempotent elements (we compute the ranks)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(x - 1)^4 * x^8,\n",
       " x^4 * (x - 1)^8,\n",
       " x^4 * (x - 1)^8,\n",
       " x^12,\n",
       " (x - 1)^12,\n",
       " (x - 1)^4 * x^8,\n",
       " (x - 1)^4 * x^8,\n",
       " x^4 * (x - 1)^8]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E=[matrix(12,12,c) for c in [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [-1, 0, -1, -1, -1, -1, 0, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, -1, 0, 0, -1, 1, 0, -1, 0, -1, 0, 0, 0, -1, 1, 0, -2, 3, 2, -1, 0, -1, 1, 1, 1, 0, 1, -1, 1, -1, -1, 0, 0, 1, 0, 0, -1, -1, 0, 1, 1, -1, -1, 1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, -1, 1, 0, -1, 0, -1, -2, 1, 1, 0, 0, 1, 2, 0, 1, 0, 0, 1, 3, -1, -1, 0, -1, 0, 0, -2, -1, -1, -1, 0, 2, -1, -1, 1, -2, 0, 0, 1, 1, -1, 0, -1, -1, 0, 1, -1, 0, 2, -1, 0, -1, -1, 0, 0, 0, 0, 0, 0, -1, 1, -1, 1, 0, -1, 0, -1, -1, 1, 0], [-2, 2, 2, 0, 0, 0, 0, 0, 1, 2, 0, -1, -1, 2, 2, 0, -1, 0, 0, 0, 1, 3, 0, -1, -1, -1, 0, 0, 0, -1, -1, -1, 0, 1, -1, -1, -1, -3, -3, 0, -1, -1, -2, -3, -1, 0, -1, 0, -1, 1, 2, 0, 0, 0, 0, 0, 1, 1, 0, -1, 0, 2, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, -3, -1, 0, 0, -1, -2, -1, -1, -1, -1, 0, 3, 0, -1, -1, 0, 1, 2, 2, -1, -3, 1, 2, 1, 1, 0, 0, 3, 2, 2, 2, 0, -3, 2, 1, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 1, 0, 1, 0, -1, 0, -1, 0, 1, 1, -1, -2, 0, -1, 0, 0, 0, -1, -1, 0, 0]]\n",
    "\n",
    "_.<a,b,c>=QQ[]\n",
    "A=a*E[0]+b*E[1]+c*E[2]\n",
    "[(s[a]*E[0]+s[b]*E[1]+s[c]*E[2]).charpoly().factor() for s in ideal((A^2-A).list()).variety()]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "A little more modest example: a bielliptic genus $2$ curve. We can check that its jacobian is isogenous to a product of elliptic curves:\n",
    "$$C\\colon y^2=x^6+x^4-5x^2+2$$\n",
    "$$E_1\\colon y^2=x^3+x^2-5x+2$$\n",
    "$$E_2\\colon y^2=1+x-5x^2+2x^3$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 1.18 s, sys: 966 µs, total: 1.18 s\n",
      "Wall time: 1.18 s\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
       "0 & 1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & -1 \\\\\n",
       "0 & 0 & 0 & 1 \\\\\n",
       "0 & 1 & 0 & 1\n",
       "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
       "-2 & 0 & -1 & 0 \\\\\n",
       "1 & 0 & 0 & 0 \\\\\n",
       "1 & 0 & 0 & 0 \\\\\n",
       "1 & 0 & 1 & 0\n",
       "\\end{array}\\right)\\right]</script></html>"
      ],
      "text/plain": [
       "[\n",
       "[ 0  1  0  0]  [-2  0 -1  0]\n",
       "[ 0  0  0 -1]  [ 1  0  0  0]\n",
       "[ 0  0  0  1]  [ 1  0  0  0]\n",
       "[ 0  1  0  1], [ 1  0  1  0]\n",
       "]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f1=y^2-(x^6+x^4-5*x^2+2)\n",
    "f2=y^2-(x^3+x^2-5*x+2)\n",
    "f3=y^2-(1+x-5*x^2+2*x^3)\n",
    "S=RiemannSurface(f1)\n",
    "E1=RiemannSurface(f2)\n",
    "E2=RiemannSurface(f3)\n",
    "%time A=S.isogeny_basis(E1+E2);\n",
    "show(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "f1=x^4*y - 2*x^2*y^3 - 3*y^5 + x^4 - 6*x^2*y^2 + 9*y^4 - 4\n",
    "f2=y^2-(x^6-x^5+1)\n",
    "f3=y^2-(2*x^5 + 27*x^4 - 54*x^2 + 27)\n",
    "S1=RiemannSurface(f1)\n",
    "S2=RiemannSurface(f2)\n",
    "S3=RiemannSurface(f3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 7.51 s, sys: 46.7 ms, total: 7.56 s\n",
      "Wall time: 7.56 s\n"
     ]
    }
   ],
   "source": [
    "%time E=S1.isogeny_basis(S2+S2+S3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{4, 8, 12}"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{sum(e).rank() for e in subsets(E) if len(e)>0}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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