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 Uq(gl(1|1)). Specifically, we consider two subalgebras Cr (n, S) and Cl (n, S) of Ozsváth-Szabó‘s algebra B(n, S), and identify their Grothendieck groups with tensor products of representations V and V* of Uq(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 givenmulti-Alexander gradings, we demonstrate a relationship between the decategorification of Ozsváth-Szabó‘s theory and Viro’s quantum relative A1 of the Reshetikhin-Turaev functor based on Uq(gl(1|1)).