Conjectured enumeration of Vassiliev invariants

A rational Ansatz is proposed for the generating function ∑ j,k β2j+k,2jx y, where βm,u is the number of primitive chinese character diagrams with u univalent and 2m− u trivalent vertices. For Pm := ∑ u≥2 βm,u, the conjecture leads to the sequence 1, 1, 1, 2, 3, 5, 8, 12, 18, 27, 39, 55, 78, 108, 150, 207, 284, 388, 532, 726 for primitive chord diagrams of degrees m ≤ 20, with predictions underlined. The asymptotic behaviour limm→∞ Pm/r m = 1.06260548918755 results, with r = 1.38027756909761 solving r = r + 1. Vassiliev invariants of knots are then enumerated by 0, 1, 1, 3, 4, 9, 14, 27, 44, 80, 132, 232, 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, 932, 1591, 2686, 4537, 7602, 12730, 21191, 35222 These conjectures are motivated by successful enumerations of irreducible Euler sums. Predictions for β15,10, β16,12 and β19,16 suggest that the action of sl and osp Lie algebras, on baguette diagrams with ladder insertions, fails to detect an invariant in each case. ) [email protected]; http://physics.open.ac.uk/ d̃broadhu