Equivariant Jones Polynomials of periodic links

TL;DR

This paper explores equivariant Jones polynomials for periodic links, establishing their properties, skein relations, and state-sum formulas, thereby enhancing periodicity criteria and revalidating classical congruences.

Contribution

It introduces the equivariant Jones polynomial, proves its skein relation, and develops a state-sum formula, advancing the understanding of periodic links.

Findings

Equivariant Jones polynomials satisfy a skein relation.

The state-sum formula enables reproving classical congruences.

Strengthens periodicity criteria for links.

Abstract

This paper continues the study of periodic links started in \cite{Politarczyk2}. It contains a study of the equivariant analogues of the Jones polynomial, which can be obtained from the equivariant Khovanov homology. In this paper we describe basic properties of such polynomials, show that they satisfy an analogue of the skein relation and develop a state-sum formula. The skein relation in the equivariant case is used to strengthen the periodicity criterion of Przytycki from \cite{Przytycki}. The state-sum formula is used to reproved the classical congruence of Murasugi from \cite{Murasugi1}.