A combinatorial one-cocycle in a moduli space of knots from the Vassiliev invariant of order 3

TL;DR

This paper proves that a Vassiliev invariant of order 3 defines a non-trivial 1-cocycle in the moduli space of knots, expanding the understanding of knot invariants via combinatorial and topological methods.

Contribution

It establishes that a specific Vassiliev invariant of order 3 indeed forms a 1-cocycle in the knot moduli space, confirming a conjecture from previous work.

Findings

Confirmed the Vassiliev invariant $v_3$ as a 1-cocycle

Extended combinatorial methods to new knot invariants

Provided a proof using established techniques from prior research

Abstract

The theory of Gauss diagrams and Gauss diagram formulas provides convenient ways to compute knot invariants, such as coefficients of the HOMFLYPT polynomial. In \cite{4,5}, the author uses Gauss diagram formulas to find combinatorial 1-cocycles in the moduli space of knots in the solid torus. Evaluated on canonical loops, one can then obtain new, non trivial knot invariants. In those books, the author conjectures that a new formula, based on the Vassiliev invariant $v_3$ also gives a 1-cocycle. We prove that it is in fact true by using the same methods developed by the author in those books.