dorsal/arxiv
View SchemaConjectured enumeration of Vassiliev invariants
| Authors | D. J. Broadhurst |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9709031 |
| URL | https://arxiv.org/abs/q-alg/9709031 |
Abstract
A rational Ansatz is proposed for the generating function $\sum_{j,k} \beta_{2j+k,2j}x^j y^k$, where $\beta_{m,u}$ is the number of primitive chinese character diagrams with $u$ univalent and $2m-u$ trivalent vertices. For $P_m:=\sum_{u\ge2}\beta_{m,u}$, the conjecture leads to the sequence $$1,1,1,2,3,5,8,12,18,27,39,55,\underline{78,108,150,207,284,388,532,726}$$ for primitive chord diagrams of degrees $m\le20$, with predictions underlined. The asymptotic behaviour $\lim_{m\to\infty}P_m/r^m= 1.06260548918755$ results, with $r=1.38027756909761$ solving $r^4=r^3+1$. Vassiliev invariants of knots are then enumerated by $$0,1,1,3,4,9,14,27,44, 80,132,232,\underline{384,659,1095,1851,3065,5128,8461,14031}$$ and Vassiliev invariants of framed knots by $$1,2,3,6,10,19,33,60,104,184,316, 548,\underline{932,1591,2686,4537,7602,12730,21191,35222}$$ These conjectures are motivated by successful enumerations of irreducible Euler sums. Predictions for $\beta_{15,10}$, $\beta_{16,12}$ and $\beta_{19,16}$ suggest that the action of sl and osp Lie algebras, on baguette diagrams with ladder insertions, fails to detect an invariant in each case.
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"date_created": "2026-03-02T18:01:28.414000Z",
"date_modified": "2026-03-02T18:01:28.414000Z",
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"abstract": "A rational Ansatz is proposed for the generating function $\\sum_{j,k}\n\\beta_{2j+k,2j}x^j y^k$, where $\\beta_{m,u}$ is the number of primitive chinese\ncharacter diagrams with $u$ univalent and $2m-u$ trivalent vertices. For\n$P_m:=\\sum_{u\\ge2}\\beta_{m,u}$, the conjecture leads to the sequence\n$$1,1,1,2,3,5,8,12,18,27,39,55,\\underline{78,108,150,207,284,388,532,726}$$ for\nprimitive chord diagrams of degrees $m\\le20$, with predictions underlined. The\nasymptotic behaviour $\\lim_{m\\to\\infty}P_m/r^m= 1.06260548918755$ results, with\n$r=1.38027756909761$ solving $r^4=r^3+1$. Vassiliev invariants of knots are\nthen enumerated by $$0,1,1,3,4,9,14,27,44,\n80,132,232,\\underline{384,659,1095,1851,3065,5128,8461,14031}$$ and Vassiliev\ninvariants of framed knots by $$1,2,3,6,10,19,33,60,104,184,316,\n548,\\underline{932,1591,2686,4537,7602,12730,21191,35222}$$ These conjectures\nare motivated by successful enumerations of irreducible Euler sums. Predictions\nfor $\\beta_{15,10}$, $\\beta_{16,12}$ and $\\beta_{19,16}$ suggest that the\naction of sl and osp Lie algebras, on baguette diagrams with ladder insertions,\nfails to detect an invariant in each case.",
"arxiv_id": "q-alg/9709031",
"authors": [
"D. J. Broadhurst"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "Conjectured enumeration of Vassiliev invariants",
"url": "https://arxiv.org/abs/q-alg/9709031"
},
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