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Description

The Alexander polynomial for knots and links can be interpreted as a quantum knot invariant associated with the quantum group of the Lie superalgebra $\mathfrak{gl}(1|1)$. This polynomial has been famously categorified to a link homology theory, knot Floer homology, defined within the theory of Heegaard-Floer homology. Andy Manion showed that the Ozsvath-Szabo algebras used to efficiently compute knot Floer homology categorify certain tensor products of $\mathfrak{gl}(1|1)$ representations. For representation theorists, the work of Sartori provides a different categorification of these same tensors products using subquotients of BGG category $\mathcal{O}$. In this talk we will explain joint work with Andy Manion establishing a direct relationship between these two constructions. Given the radically different nature of these two constructions, transporting ideas between them provides a new perspective and allows for new results that would not have been apparent otherwise.