On the decategorification of Ozsv\'ath and Szab\'o's bordered theory for knot Floer homology

Abstract

We relate decategorifications of Ozsv\'ath-Szab\'o'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 Ozsv\'ath- Szab\'o's algebra $\mathcal{B}(n,\mathcal{S})$, and identify their Grothendieck groups with tensor products of representations $V$ and $V^*$ of $\mathcal{U}_q(\mathfrak{gl}(1|1))$, where $V$ is the vector representation. We identify the decategorifications of Ozsv\'ath-Szab\'o's DA bimodules for elementary tangles with corresponding maps between representations. Finally, when the algebras are given multi-Alexander gradings, we demonstrate a relationship between the decategorification of Ozsv\'ath-Szab\'o's theory and Viro's quantum relative $\mathcal{A}^1$ of the Reshetikhin-Turaev functor based on $\mathcal{U}_q(\mathfrak{gl}(1|1))$.