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Quantum Topology


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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