Vassiliev invariants for knots from the ADO polynomials

TL;DR

This paper establishes a connection between ADO polynomials and Vassiliev invariants for knots, providing new methods for their expansion and analyzing their asymptotic behavior as the parameter grows large.

Contribution

It introduces a way to expand ADO polynomials as Vassiliev invariants, and relates them to colored Jones polynomials through a unified invariant.

Findings

ADO polynomials can be expanded as Vassiliev invariants for prime power r

A unique, computable expansion yields r-adic topological Vassiliev invariants

Asymptotic behavior of ADO polynomials modulo r analyzed as r approaches infinity

Abstract

In this paper we prove that the $r$-th ADO polynomial of a knot, for $r$ a power of prime number, can be expanded as Vassiliev invariants with values in $\mathbb{Z}$. Nevertheless this expansion is not unique and not easily computable. We can obtain a unique computable expansion, but we only get $r$ adic topological Vassiliev invariants as coefficients. To do so, we exploit the fact that the colored Jones polynomials can be decomposed as Vassiliev invariants and we tranpose it to ADO using the unified knot invariant recovering both ADO and colored Jones defined in arXiv:2003.09854. Finally we prove some asymptotic behavior of the ADO polynomials modulo $r$ as $r$ goes to infinity.