Khovanov homology for signed divides

Abstract

The purpose of this paper is to interpret polynomial invariants of strongly invertible links in terms of Khovanov homology theory. To a divide, that is a proper generic immersion of a finite number of copies of the unit interval and circles in a $2$–disc, one can associate a strongly invertible link in the $3$–sphere. This can be generalized to signed divides: divides with $+$ or $-$ sign assignment to each crossing point. Conversely, to any link $L$ that is strongly invertible for an involution $j$, one can associate a signed divide. Two strongly invertible links that are isotopic through an isotopy respecting the involution are called strongly equivalent. Such isotopies give rise to moves on divides. In a previous paper [Topology 47 (2008) 316-350], the author finds an exhaustive list of moves that preserves strong equivalence, together with a polynomial invariant for these moves, giving therefore an invariant for strong equivalence of the associated strongly invertible links. We prove in this paper that this polynomial can be seen as the graded Euler characteristic of a graded complex of ${\mathbb{Z}}_2$–vector spaces. Homology of such complexes is invariant for the moves on divides and so is invariant through strong equivalence of strongly invertible links.