34:["$","title",null,{"children":"Computing homology basis of algebraic curves, TT84 algorithm. | pascalmolin | Observable"}] 3c:T869,fullgraph = { const rx = 100, ry = 40, y0 = 30, wy = 30; const svg = d3.select(DOM.svg(width, width)) .style("width", "100%") .style("height", "auto") .style("padding", "10px") .style("box-sizing", "border-box") .style("font", "10px sans-serif"); const g = svg.append("g"); const curve = g.append("g") .attr("fill", "none") .attr("stroke", "#555") //.attr("stroke-opacity", 0.4) .attr("stroke-width", .5) //.attr("transform", `translate(${y0},0)`) ; function px(v) { return v instanceof Sheet ? 0 : rx * Math.cos(.17+v.index*6.28/(n+d)); } function py(v) { return v instanceof Sheet ? v.index * wy - y0 : ry * Math.sin(.17+v.index*6.28/(n+d)); } const graph = d3.merge(branchpoints.map(p=>p.connected().map(s=>({a:p,b:s})))) const col = d3.scaleOrdinal(d3.schemeSet2) //return graph; const edges = curve.append("g") .attr("stroke-opacity", 0.7) .attr("stroke-width", 1) .selectAll("line") .data(graph) .enter().append("line") .attr("x1", v=>px(v.a)) .attr("y1",v=>py(v.a)) .attr("x2", 0) .attr("y2",v=>py(v.b)) .attr("stroke", v=>col(v.b.index)) ; const vertices = curve .append("g") .selectAll("g") .data(branchpoints) .enter().append("g") .attr("transform", d => `translate(${px(d)},${py(d)})`); vertices.append("circle") .attr("fill", "#444") .attr("r", 1); vertices.append("text") .attr("dy", "0.31em") .attr("dx", "0.31em") .text(d => d.name); const sheets = curve .append("g") .selectAll("g") .data(y) .enter().append("g") .attr("transform", d => `translate(0,${py(d)})`) .attr("stroke", d=>col(d.index)) ; sheets.append("circle") .attr("r", 1); sheets.append("text") .attr("dy", "-0.31em") .attr("dx", "-0.31em") .text(d => d.name); document.body.appendChild(svg.node()); const box = g.node().getBBox(); svg.remove() .attr("width", box.width) .attr("height", box.height) .attr("viewBox", `${box.x} ${box.y} ${box.width} ${box.height}`); return svg.node(); }3d:T8eb,graphtree = { const h = d3.hierarchy(tree) //return h; const root = d3.tree() .size([2 * Math.PI, radius]) .separation((a, b) => (a.parent == b.parent ? 1 : 2) / a.depth) (h); //return root.links(); const svg = d3.select(DOM.svg(width, width)) .style("width", "50%") .style("height", "auto") .style("padding", "10px") .style("box-sizing", "border-box") .style("font", "10px sans-serif"); const g = svg.append("g"); const link = g.append("g") .attr("fill", "none") .attr("stroke", "#555") .attr("stroke-opacity", 0.4) .attr("stroke-width", 1.5) .selectAll("path") .data(root.links()) .enter() .append("path") .attr("stroke-opacity", d => d.target.data.count < steps ? 0.7 : 0) .attr("stroke", d => d.target.data.count == steps-1 ? 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while(b) { var t = b ; b = a % b; a = t}; return a } ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1565,"value":"md`## Sheets and branch points`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":426,"value":"md`We introduce the m sheets`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1017,"value":"function Sheet(i,name) { this.index = i; this.name = name}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":428,"value":"y = Array(m).fill(0).map((x,i)=> new Sheet(i,\"y\"+(i+1)))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":416,"value":"md`and the branch points ${tex `x_i`}. They are connected to all sheets and their monodromy is the cycle ${tex `(1,2,\\dots m)`}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":937,"value":"function BranchPoint(i,name,mon, sheets) { this.index = i; this.name = name; this.mon = i=>(i+mon)%m; this.connected = x=>sheets }","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":408,"value":"x = Array(n).fill(0).map((x,i)=> new BranchPoint(i,\"x\"+(i+1),1,y))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":393,"value":"md`\nThe d points at infinity correspond to the cycles of the inverse permutation of\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":438,"value":"tex`(1,2,\\dots m)^n = (1,n+1,2n+1,\\dots (m/g-1)n+1)(2,n+2,\\dots)(g,n+g,\\dots (m/g-1)n+g)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":446,"value":"md`\n(this is so that the total product of all monodromies is the identity).\nThe point ${tex `\\infty_i`} is connected to the sheets ${tex `y_i, y_{d+i},\\dots`}.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":422,"value":"oo = Array(d).fill(0).map((x,i)=> new BranchPoint(n+i,\"oo\"+(i+1),(m*n)-n, y.filter(v => (v.index % d) == i)))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1562,"value":"md`In fact, ${tex `\\infty`}-points are branch points iff ${tex `d p.connected().includes(this) )) }","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1388,"value":"md`For example the point ${tex `\\infty_1`} is connected to the following sheets`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1112,"value":"oo[0].from(y[0])","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":451,"value":"md`## A complex\nWe now consider the full graph ${tex `G`} drawn on the curve, whoses vertices are the sheets and branch points, and edges connect sheets to the branch points connected to them.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":453,"value":"$3c","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":570,"value":"md`\n## Covering tree and homology basis\nOur goal is to compute a basis of the homology of C. By homotopy, any cycle can be deformed into a path on the full graph above. So we are going to search for all closed path starting from one basepoint, say the point y1 on the first sheet.\n\nFrom a breadth-first traversal of the graph, we contruct a covering tree, stopping a path when it reaches a vertex already visited. Paths cannot pass twice the same vertex.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":675,"value":"$3d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1275,"value":"$3e","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":572,"value":"viewof steps = html``\n/*steps = {\n var max = 2*((n-1)*(m-1)+2*d)+2*m+5\n for (let i = 0;;i=0)\n while(i < max)\n yield Promises.delay(1000, i++);\n}*/","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1524,"value":"md`\nWe have shown two things in this tree:\n- the fat subtree forms a maximal spanning tree in the complete graph\n- the next thin edges join an already visited vertex, making loops on the curve.\n\nEach thin edge on the graph appears twice on the planar tree. This corresponds to traversing the edge of the graph once in each direction.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1571,"value":"md`## Homology basis\n\nIf we take a path along the tree from the middle of one thin edge to the middle of its sibling, it corresponds to a closed loop on the curve.\n\nWe now claim:\n1. those loops form a homology basis of the graph G.\n\nIn fact, contracting the fat maximal tree into a single vertex on the curve we obtain a bouquet of N circles, where N is the number of thin edges on the graph. Hence N=2g+2 (=${(m-1)*(n-1)+1-d+2} here).\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1537,"value":"md`## The intersections\n\nWe also have\n2. we can build the tree in such a way that the intersection number of two such loops on the curve is the intersection of the paths on the planar tree.\n\nIn order to have the second claim, we need to arrange the planar tree so that it preserves orientation at branch points. We do this by always disposing the edges around a branch point in the order given by the action of the local monodromy.\n\n\nNow we have a method to compute the intersection pairing. If we have two paths in the tree, each joining sibling edges, then their intersection number is also the intersection of the lines joining the end-points, and we can arrange all these end-points to be on a circle.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1324,"value":"$3f","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1558,"value":"md`## Other strategies`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1234,"value":"viewof algorithm = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1371,"value":"md`## The traversal algorithm\n\nWe define nodes, which can be Sheets or BranchPoints.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":810,"value":"class Node {\n constructor(v,p) {\n this.value = v;\n this.children = [];\n this.parent = p;\n //this.ancestors = p ? p.ancestors.concat(v) : [v] // FIXME: remove?\n }\n has_ancestor(x) {\n return (this.value == x || (this.parent && this.parent.has_ancestor(x)))\n }\n new_child(x) {\n var n = new Node(x,this)\n this.children.push(n)\n return n\n }\n connected(from) {\n return this.value.from(from ? from.value : null)\n }\n descendants () {\n var l = [this]\n this.children.forEach(x=>l = l.concat(x.descendants()))\n return l\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":797,"value":"function breadth_traversal(v) {\n var count = 0, visited = [], leafs = [];\n var root = new Node(v,null);\n var todo = [ root ];\n while(todo.length)\n {\n var tmp = []\n for (let v of todo)\n {\n v.connected(v.parent)\n .filter(x => !v.has_ancestor(x))\n .forEach( function(x) {\n var n = v.new_child(x);\n n.count = count++;\n if (visited.includes(x))\n {\n leafs.push(n);\n }\n else\n {\n visited.push(x);\n tmp.push(n);\n }\n });\n todo = tmp;\n }\n }\n return root;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1265,"value":"function depth_traversal(v) {\n var count = 0, visited = [], leafs = []\n var root = new Node(v,null);\n var todo = [ root ]\n while(todo.length)\n {\n let v = todo.pop()\n let l = v.connected(v.parent)\n .filter(x => !v.has_ancestor(x))\n .map( x => v.new_child(x) )\n \n l.forEach( function(n) {\n var n = v.new_child(x);\n n.count = count++;\n if (visited.includes(x))\n {\n leafs.push(n);\n }\n else\n {\n visited.push(x);\n todo.push(n);\n }\n });\n }\n return root;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1158,"value":"md`\nAs explained above, we must dispose the children of a node according to the monodromy, and implement the descent iterator accordingly.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1054,"value":"Sheet.prototype.from = function(point) { return this.connected() }","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1050,"value":"BranchPoint.prototype.from = function(sheet) {\n if (!sheet)\n {\n return this.connected();\n }\n else\n {\n var s0 = sheet.index, s = this.mon(s0), l = [];\n while(s != s0)\n {\n l.push(y[s]);\n s = this.mon(s);\n }\n return l;\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":776,"value":"tree = algorithm == \"breadth\" ? breadth_traversal(y[0]) : depth_traversal(y[0])","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1611,"value":"md`## More general curve\n\nLet ${tex `f = xy^4+yx^2+1`}. The function x ramifies with index 4 above 0, with index 2 above 7 points\nx1,...x7, and with index 3 above oo.\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1615,"value":"Y = Array(4).fill(0).map((x,i)=> new Sheet(i,\"Y\"+(i+1)))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1621,"value":"function local_mon(cycle) {\n return function (i) { var z; return (z = cycle.indexOf(i)>=0) ? cycle[z+1] : null; };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1626,"value":"function RamPoint(i,name, cycle, sheets) {\n var ls = cycle.map(i=>sheets[i]);\n this.index = i; this.name = name;\n this.cycle = cycle;\n this.mon = local_mon(cycle);\n this.connected = x=>ls;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1617,"value":"X = [ new RamPoint(0,\"0\",[0,1,3,2],Y),\n new RamPoint(1,\"X1\",[0,1],Y),\n new RamPoint(2,\"X2\",[0,1],Y),\n new RamPoint(3,\"X3\",[0,2],Y),\n new RamPoint(4,\"X4\",[2,3],Y),\n new RamPoint(5,\"X5\",[2,3],Y),\n new RamPoint(6,\"X6\",[0,2],Y),\n new RamPoint(7,\"X7\",[1,3],Y),\n new RamPoint(8,\"oo\",[1,3,2],Y)\n ]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1624,"value":"Y[0].connected = ()=>X.filter(x=>x.cycle.contains(0))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1651,"value":"Y[1].connected = ()=>X.filter(x=>x.cycle.contains(1))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1683,"value":"Y[2].connected = ()=>X.filter(x=>x.cycle.contains(2))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1685,"value":"Y[3].connected = ()=>X.filter(x=>x.cycle.contains(3))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1687,"value":"T2 = breadth_traversal(Y[0])","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1555,"value":"md`# Extra stuff, unused`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":873,"value":"function Point(index) {\n this.index = index;\n this.name = \"y\"+(index+1);\n this.connected = function (l) {\n if (l)\n { \n this._connected = l;\n return this;\n } \n else\n return this._connected;\n }\n this.connected_from = function (i) {\n return this._connected.slice(i+1).concat(this._connected.slice(0,i))\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":875,"value":"P = Array(5).fill(0).map((i,x)=>new Point(x,i))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":884,"value":"P[0].connected([P[1],P[2]]),\nP[2].connected([P[1],P[4]]),\nP[3].connected([P[0],P[2]]),\nP[4].connected([P[1],P[3]])","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":104,"value":"$40","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":99,"value":"width = 500//932","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":101,"value":"radius = width / 2","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":7,"value":"d3 = require(\"d3@5\")","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W41","stdlib":"1"}} 33:null