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

TL;DR

This paper connects Ozsváth-Szabó's bordered knot Floer homology decategorification to quantum group representations, revealing algebraic structures and maps that relate to Viro's quantum invariants.

Contribution

It establishes a link between decategorified bordered knot Floer theory and representations of quantum superalgebra U_q(gl(1|1)), including explicit algebraic and functorial correspondences.

Findings

Grothendieck groups identified with tensor products of U_q(gl(1|1)) representations

Decategorifications of DA bimodules correspond to maps between these representations

Relationship established between decategorification and Viro's quantum relative A^1

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 $\mathcal{U}_q(\mathfrak{gl}(1|1))$.