Khovanov homology of graph-links

Abstract

We define Khovanov homology over Z2 for graph-links.

Unsolved problems in virtual knot theory and combinatorial knot theory

“…Given two diagrams of (virtual) knots or links with exactly one circle in the A-state. [11,193,194] approach to Khovanov homology [195,196] can be used here. 41.…”

“…The equivalence of the two theories (the theory of looped interlacement graphs and the theory of graph-knots) was proved in [67]. Also, some invariants were constructed, see [68,69,71,74,193,213]. 1.…”

“…It is worth comparing these thoughts with the recent work of Lipschitz on the spectrification of Khovanov homology. Possibly, the Bloom-Nikonov [11,193,194] approach to Khovanov homology [195,196] can be used here.…”

Parity in knot theory and graph-links

“…Note that the theory of Khovanov homology is constructed for graph-knots and, therefore, for homotopy classes of looped interlacement graphs [183,184]. The Khovanov homology will be considered in Sec.…”

Odd Khovanov homology of principally unimodular bipartite graph-links

“…This article is a sequel of our paper [6] and describes the integral version of odd Khovanov homology for graph-links. We refer the reader to papers [4,5] for the definition of graph-links and to papers [1,7] for the construction of odd Khovanov homology.…”