The Knot Floer Cube of Resolutions and the Composition Product

TL;DR

This paper explores the connection between knot Floer homology and HOMFLY-PT homology, revealing a decomposition that relates to the HOMFLY-PT polynomial through a filtration and composition product.

Contribution

It introduces a filtration on the knot Floer cube of resolutions that decomposes into HOMFLY-PT homologies, linking these invariants via Jaeger's composition product formula.

Findings

Filtered complex decomposes into HOMFLY-PT homologies of subdiagrams

Graded Euler characteristic matches the HOMFLY-PT polynomial

Establishes a new relationship between knot invariants

Abstract

We examine the relationship between the (untwisted) knot Floer cube of resolutions and HOMFLY-PT homology. By using a filtration induced by additional basepoints on the Heegaard diagram for a knot $K$, we see that the filtered complex decomposes as a direct sum of HOMFLY-PT homologies of various subdiagrams. Jaeger's composition product formula shows that the graded Euler characteristic of this direct sum is the HOMFLY-PT polynomial of $K$.