Odd Vassiliev invariants vanish at four loops

Abstract

The odd Vassiliev conjecture asserts that all Jacobi diagrams with an odd number of legs vanish modulo the AS and IHX relations. Equivalently, the conjecture asserts that Vassiliev invariants cannot distinguish a knot from its inverse. The conjecture is elementary for one 1 or 2 loops, and was proven by Moskovich and Ohtsuki for 3 loops. We establish the conjecture for 4 loops. Our proof uses Jacobi diagrams and weight systems with polynomial coefficients, commuting colored legs, and several vanishing criteria.