Link Floer homology categorifies the Conway function

TL;DR

This paper proves that the Euler characteristic of link Floer homology exactly equals the Conway function, providing a precise categorification of the multivariate Alexander polynomial for links.

Contribution

It establishes that the link Floer homology's Euler characteristic matches the Conway function, removing previous ambiguities and connecting Floer homology with classical link invariants.

Findings

Euler characteristic of link Floer homology equals the Conway function

Develops a model of the Conway function for rectangular diagrams

Connects combinatorial Floer homology with classical link invariants

Abstract

Given an oriented link in the 3-sphere, the Euler characteristic of its link Floer homology is known to coincide with its multivariate Alexander polynomial, an invariant only defined up to a sign and powers of the variables. In this paper, we get rid of this ambiguity by proving that this Euler characteristic is equal to the so-called Conway function, the representative of the multivariate Alexander polynomial introduced by Conway in 1970 and explicitly constructed by Hartley in 1983. This is achieved by creating a model of the Conway function adapted to rectangular diagrams, which is then compared to the Euler characteristic of the combinatorial version of link Floer homology.