A link invariant related to Khovanov homology and knot Floer homology

Akram Alishahi; Nathan Dowlin

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arXiv · 1810.13406

A link invariant related to Khovanov homology and knot Floer homology

Akram Alishahi·Nathan Dowlin

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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.

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Akram Alishahi, Nathan Dowlin. 2018-10-31. A link invariant related to Khovanov homology and knot Floer homology. https://arxiv.org/abs/1810.13406

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