Equivariant Jones polynomials of periodic links

Abstract

This paper studies equivariant Jones polynomials of periodic links. Namely, to every [Formula: see text]-periodic link and any divisor [Formula: see text] of [Formula: see text], we associate a polynomial that is a graded Euler characteristic of [Formula: see text]-graded equivariant Khovanov homology. The first main result shows that certain linear combinations of these polynomials, called the difference Jones polynomials, satisfy an appropriate version of the skein relation. This relation is used to generalize Przytycki’s periodicity criterion. We also provide an example showing that the new criterion is indeed stronger. The second main result gives a state-sum formula for the difference Jones polynomials. This formula is used to give an alternative proof of the periodicity criterion of Murasugi.