The Khovanov Homology of Alternating Virtual Links

Abstract

In this paper, we study the Khovanov homology of an alternating virtual link L and show that it is supported on $\mathit{g}+2$ diagonal lines, where g equals the virtual genus of L. Specifically, we show that $\mathit{K}{\mathit{h}}^{\mathit{i},\mathit{j}}(\mathit{L})$ is supported on the lines $\mathit{j}=2\mathit{i}-{\mathit{\sigma }}_{\mathit{\xi }}+2\mathit{k}-1$ for $0\le \mathit{k}\le \mathit{g}+1$ where ${\mathit{\sigma }}_{{\mathit{\xi }}^{\ast }}(\mathit{L})+2\mathit{g}={\mathit{\sigma }}_{\mathit{\xi }}(\mathit{L})$ are the signatures of L for a checkerboard coloring ξ and its dual ${\mathit{\xi }}^{\ast }$. Of course, for classical links, the two signatures are equal, and this recovers Lee’s H-thinness result for $\mathit{K}{\mathit{h}}^{\ast ,\ast }(\mathit{L})$. Our result applies more generally to give an upper bound for the homological width of the Khovanov homology of any checkerboard virtual link L. The bound is given in terms of the alternating genus of L, which can be viewed as the virtual analogue of the Turaev genus. The proof rests on associating, with any checkerboard colorable link L, an alternating virtual link diagram with the same Khovanov homology as L.

In the process, we study the behavior of the signature invariants under vertical and horizontal mirror symmetry. We also compute the Khovanov homology and Rasmussen invariants in numerous cases and apply them to show nonsliceness and determine the slice genus for several virtual knots. ??Table]ras-table at the end of the paper lists the signatures, Khovanov polynomial, and Rasmussen invariant for alternating virtual knots up to six crossings.