Evaluations of Link Polynomials and Recent Constructions in Heegaard Floer Theory

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 $HFK_n$.