The number of primitive Vassiliev invariants up to degree 12

TL;DR

This paper computes the number of primitive Vassiliev invariants up to degree 12, revealing their structure and disproving a conjecture, with implications for knot theory and Lie algebra representations.

Contribution

It provides the first exact counts for degrees 10 to 12 and shows all invariants up to degree 12 originate from Lie algebra representations, also falsifying Vogel's conjecture.

Findings

Exact counts for primitive Vassiliev invariants up to degree 12

All invariants below degree 13 are orientation insensitive

Vogel's conjecture is falsified

Abstract

We present algorithms giving upper and lower bounds for the number of independent primitive rational Vassiliev invariants of degree m modulo those of degree m-1. The values have been calculated for the formerly unknown degrees m = 10, 11, 12. Upper and lower bounds coincide, which reveals that all Vassiliev invariants of degree smaller 13 are orientation insensitive and are coming from representations of Lie algebras so and gl. Furthermore, a conjecture of Vogel is falsified and it is shown that the \Lambda-module of connected trivalent diagrams (Chinese characters) is not free.