Quantum gl(1|1) and tangle Floer homology

TL;DR

This paper establishes a connection between tangle Floer homology and quantum superalgebra representations, showing that certain algebraic structures correspond to topological tangle invariants.

Contribution

It identifies the Grothendieck group of tangle Floer dg algebra with tensor products of $U_q(gl(1|1))$ representations and relates bimodule maps to Reshetikhin-Turaev invariants.

Findings

Grothendieck group corresponds to $U_q(gl(1|1))$ representations

Bimodule maps match Reshetikhin-Turaev homomorphisms

Introduces dg bimodules acting as algebra generators

Abstract

We identify the Grothendieck group of the tangle Floer dg algebra with a tensor product of certain $U_q(gl(1|1))$ representations. Under this identification, up to a scalar factor, the map on the Grothendieck group induced by the tangle Floer dg bimodule associated to a tangle agrees with the Reshetikhin-Turaev homomorphism for that tangle. We also introduce dg bimodules which act on the Grothendieck group as the generators $E$ and $F$ of $U_q(gl(1|1))$.