The EMS Publishing House is now EMS Press and has its new home at

Please find all EMS Press journals and articles on the new platform.

Quantum Topology

Full-Text PDF (1116 KB) | Metadata | Table of Contents | QT summary
Online access to the full text of Quantum Topology is restricted to the subscribers of the journal, who are encouraged to communicate their IP-address(es) to their agent or directly to the publisher at
Volume 10, Issue 1, 2019, pp. 77–206
DOI: 10.4171/QT/123

Published online: 2018-10-31

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

Andrew Manion[1]

(1) University of Southern California, Los Angeles, USA

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

Keywords: Bordered Heegaard Floer homology, Ozsváth–Szabó’s bordered theory, quantum groups, categorification

Manion Andrew: On the decategorification of Ozsváth and Szabó's bordered theory for knot Floer homology. Quantum Topol. 10 (2019), 77-206. doi: 10.4171/QT/123