Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT

TL;DR

This paper develops a method to compute link Floer homology maps induced by elementary cobordisms, establishing a connection with reduced Khovanov TQFT and introducing a spectral sequence invariant of links.

Contribution

It explicitly computes cobordism maps in link Floer homology, relates them to a TQFT equivalent to reduced Khovanov homology, and constructs a new link invariant via a spectral sequence.

Findings

Computed cobordism maps for births, deaths, stabilizations, and destabilizations.

Established the equivalence of the cobordism-induced TQFT with reduced Khovanov TQFT.

Defined a spectral sequence from Khovanov homology that is a link invariant.

Abstract

We study the maps induced on link Floer homology by elementary decorated link cobordisms. We compute these for births, deaths, stabilizations, and destabilizations, and show that saddle cobordisms can be computed in terms of maps in a decorated skein exact triangle that extends the oriented skein exact triangle in knot Floer homology. In particular, we completely determine the Alexander and Maslov grading shifts. As a corollary, we compute the maps induced by elementary cobordisms between unlinks. We show that these give rise to a $(1+1)$-dimensional TQFT that coincides with the reduced Khovanov TQFT. Hence, when applied to the cube of resolutions of a marked link diagram, it gives the complex defining the reduced Khovanov homology of the knot. Finally, we define a spectral sequence from (reduced) Khovanov homology using these cobordism maps, and we prove that it is an invariant of the (marked) link.