The purpose of this paper is to construct and study equivariant Khovanov homology -a version of Khovanov homology theory for periodic links. Since our construction works regardless of the characteristic of the coefficient ring it generalizes a previous construction by Chbili. We establish invariance under equivariant isotopies of links and study algebraic properties of integral and rational version of the homology theory. Moreover, we construct a skein spectral sequence converging to equivariant Khovanov homology and use this spectral sequence to compute, as an example, equivariant Khovanov homology of torus links T (n, 2).

Based on the results of the second author, we define an equivariant version of Lee and Bar-Natan homology for periodic links and show that there exists an equivariant spectral sequence from the equivariant Khovanov homology to equivariant Lee homology. As a result we obtain new obstructions for a link to be periodic. These obstructions generalize previous results of Przytycki and of the second author.

This paper continues the study of periodic links started in [Pol15]. 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 [Prz89]. The state-sum formula is used to reproved the classical congruence of Murasugi from [Mur88].

Given an m-periodic link $$L\subset S^3$$ L ⊂ S 3 , we show that the Khovanov spectrum $$\mathcal {X}_L$$ X L constructed by Lipshitz and Sarkar admits a group action. We relate the Borel cohomology of $$\mathcal {X}_L$$ X L to the equivariant Khovanov homology of L constructed by the second author. The action of Steenrod algebra on the cohomology of $$\mathcal {X}_L$$ X L gives an extra structure of the periodic link. Another consequence of our construction is an alternative proof of the localization formula for Khovanov homology, obtained first by Stoffregen and Zhang. By applying the Dwyer–Wilkerson theorem we express Khovanov homology of the quotient link in terms of equivariant Khovanov homology of the original link.