Conjectured enumeration of Vassiliev invariants

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 $P_m:={\sum }_{u\ge 2}{\beta }_{m,u}$, 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 $m\le 20$, 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.

10 pages, LaTeX

References in corpus (3)

• The Fundamental Theorem of Vassiliev Invariants
• The number of primitive Vassiliev invariants up to degree 12
• On the Hopf algebra structure of perturbative quantum field theories

Cited by in corpus (1)

• Problems on invariants of knots and 3-manifolds