A link invariant related to Khovanov homology and knot Floer homology

TL;DR

This paper introduces a new algebraic chain complex for knots that links Khovanov homology and knot Floer homology, providing a novel invariant with potential isomorphism to elta-graded knot Floer homology.

Contribution

It constructs a purely algebraic chain complex inspired by knot Floer homology, establishing a link to Khovanov homology and proposing a new link invariant.

Findings

The E_2 page of the spectral sequence matches Khovanov homology.

The total homology is a link invariant.

The complex can be refined to a tangle invariant with a bimodule structure.

Abstract

In this paper we introduce a chain complex $C_{1 \pm 1}(D)$ where D is a plat braid diagram for a knot K. This complex is inspired by knot Floer homology, but it the construction is purely algebraic. It is constructed as an oriented cube of resolutions with differential d=d_0+d_1. We show that the E_2 page of the associated spectral sequence is isomorphic to the Khovanov homology of K, and that the total homology is a link invariant which we conjecture is isomorphic to \delta-graded knot Floer homology. The complex can be refined to a tangle invariant for braids on 2n strands, where the associated invariant is a bimodule over an algebra A_n. We show that A_n is isomorphic to B'(2n+1, n), the algebra used for the DA-bimodule constructed by Ozsvath and Szabo in their algebraic construction of knot Floer homology.