Evaluations of link polynomials and recent constructions in Heegaard Floer theory

TL;DR

This paper demonstrates that certain Heegaard Floer knot invariants recover classical polynomial evaluations at roots of unity, linking them to $rak{sl}(n)$ polynomials and spectral sequences in knot theory.

Contribution

It introduces a new Euler characteristic definition for fractionally-graded complexes and connects Heegaard Floer invariants with $rak{sl}(n)$ polynomial evaluations at roots of unity.

Findings

Euler characteristics of $HFK_n$ recover Alexander and $rak{sl}(n)$ polynomial evaluations.

Equality of these evaluations relates to conjectured spectral sequences between $rak{sl}(n)$ homology and $HFK_n$.

Provides a decategorified perspective on the relationship between different knot invariants.

Abstract

Using a definition of Euler characteristic for fractionally-graded complexes based on roots of unity, we show that the Euler characteristics of Dowlin's "$\mathfrak{sl}(n)$-like" Heegaard Floer knot invariants $HFK_n$ recover both Alexander polynomial evaluations and $\mathfrak{sl}(n)$ polynomial evaluations at certain roots of unity for links in $S^3$. We show that the equality of these evaluations can be viewed as the decategorified content of the conjectured spectral sequences relating $\mathfrak{sl}(n)$ homology and $HFK_n$.