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# Topology Seminar – Dave Rose (UNC)

## September 24 @ 3:30 pm - 4:30 pm

**Spin link homology**

**Abstract.** For each finite-dimensional simple complex Lie algebra, Reshetikhin-Turaev define a quantum invariant of knots/links in the 3-sphere with components colored by finite-dimensional representations. These invariants generalize the Jones polynomial, which corresponds to the case of sl(2) and the vector representation. One categorical level higher, Khovanov and Khovanov-Rozansky construct sl(n) link homology theories, explicit and computable homological invariants of knots/links that “categorify” the sl(n) Reshetikhin-Turaev invariants. Extending this theory, Webster defines link homologies associated to arbitrary finite-dimensional simple complex Lie algebras. While sl(n) link homologies have been widely studied and have found spectacular applications in 3- and 4-dimensional topology, essentially nothing is known about link homologies for non type A Lie algebras, beyond their existence.

In this talk, we will present a new categorification of the spin-colored so(2n+1) link invariant, which arises from a novel involution on n-colored sl(2n) Khovanov-Rozansky homology. Our approach involves categorical representation theory, and highlights a subtle connection between link invariants in types A and B that is hidden at the decategorified level. Further, our construction is explicit and computable, thus should be amenable to topological applications. (This is joint work with Elijah Bodish and Ben Elias.)