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 $P_m:=\sum_{u\ge2}\beta_{m,u}$, 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.