Khovanov homology for signed divides

TL;DR

This paper extends Khovanov homology to signed divides, providing a homological interpretation of polynomial invariants for strongly invertible links, and demonstrating their invariance under specific moves.

Contribution

It introduces a homological framework for polynomial invariants of signed divides, linking them to Khovanov homology and invariance under moves.

Findings

Polynomial invariant is the graded Euler characteristic of a homology complex.

Homology groups are invariant under moves preserving strong equivalence.

Provides a new homological perspective on invariants of strongly invertible links.

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 of the author, one can find 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 vector spaces. Homology of such complexes is invariant for the moves on divides and so is invariant through strong equivalence of strongly invertible links.