On the decategorification of Ozsváth and Szabó's bordered theory for knot Floer homology

Abstract

We relate decategorifications of Ozsváth–Szabó's new bordered theory for knot Floer homology to representations of ${\mathcal{U}}_q(\mathfrak{gl}(1|1))$. Specifically, we consider two subalgebras ${\mathcal{C}}_r(n,\mathcal{S})$ and ${\mathcal{C}}_l(n,\mathcal{S})$ of Ozsváth–Szabó's algebra $\mathcal{B}(n,\mathcal{S})$, and identify their Grothendieck groups with tensor products of representations $V$ and $V^{\ast }$ of ${\mathcal{U}}_q(\mathfrak{gl}(1|1))$, where $V$ is the vector representation. We identify the decategorifications of Ozsváth–Szabó's $DA$ bimodules for tangles with corresponding maps between representations. Finally, when the algebras are given multi-Alexander gradings, we demonstrate a relationship between the decategorification of Ozsváth–Szabó's theory and Viro's quantum relative ${\mathcal{A}}^1$ of the Reshetikhin–Turaev functor based on ${\mathcal{U}}_q(\mathfrak{gl}(1|1))$.