Equivariant Khovanov Homology of Periodic Links

Abstract

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).

Khovanov homotopy type, periodic links and localizations

“…A diagram D of a link L is called m -periodic if and D is invariant under rotation of order m of the plane about the point . The Khovanov complex of an m -periodic link admits an induced action of [ 8 , 38 ].…”

“…The classical combinatorial definition of Khovanov homology enables an equivariant version [ 38 ], generalizing earlier constructions of Chbili [ 8 ].…”

“…Indeed, using the result of Dwyer and Wilkerson [ 14 ], it is possible to give a complete description of the Khovanov homology of the quotient knot in terms of the equivariant Khovanov homology of a p -periodic knot, for a prime p . By Theorem 8.3 , the Borel cohomology of can be identified with the equivariant Khovanov homology introduced by the second author [ 38 ]. Repeating the construction of [ 38 ] one can obtain the equivariant annular Khovanov homology , which, by an analog of Theorem 8.3 , is isomorphic to the Borel cohomology of .…”

“…By Theorem 8.3 , the Borel cohomology of can be identified with the equivariant Khovanov homology introduced by the second author [ 38 ]. Repeating the construction of [ 38 ] one can obtain the equivariant annular Khovanov homology , which, by an analog of Theorem 8.3 , is isomorphic to the Borel cohomology of . Therefore, and admit an action of the cohomology algebra , of the classifying space of and the action of the mod p Steenrod algebra .…”

A rank inequality for the annular Khovanov homology of 2–periodic links

“…The rise of homology-type invariants categorifying polynomial link invariants have led to the study of equivariant homology theories by Chbili [7], Seidel and Smith [22], Hendricks [11] [12], Politarczyk [17] [18], Borodzik and Politarczyk [6], and many others, building and improving upon earlier work of Murasugi [15] [16], Yokota [23], and Przytycki [19] relating the polynomial invariants of periodic links with those of their quotient links.…”