Local and Quasi-Local sl(2) Link Homology

Hogancamp, Matthew, Mathematics - Graduate School of Arts and Sciences, University of Virginia
Krushkal, Vyacheslav, Department of Mathematics, University of Virginia

We study algebraic structures in the sl2 link homology theories as defined by Khovanov [Kho00] and Bar-Natan [BN05] and extended by Cooper-Krushkal [CK12]. New duality results are applied to the projector P_n which governs the local behavior of these link invariants. As an application, we construct an action of the polynomial ring Z[u_1, . . . , u_n] on P_n and prove that the colored sl2 link homology is finitely generated over a tensor product of such rings. Replacing P_n by the associated Koszul complex Z ⊗_Z[u_1,...,u_n] P_n gives a categorification of the sl2 Reshetikhin- Turaev invariant (up to normalization) via bounded chain complexes. The invariant is quasi-local, i.e. extends to tangles and respects gluing up to taking many direct sum copies. We conjecture that our link invariant is functorial under link cobordisms (up to sign), and as evidence for this we show that the intermediate chain complexes Qn = Z ⊗_Z[u_n] P_n are Frobenius algebra objects in an appropriate monoidal category. Combining these results allows us to simplify the endomorphism algebra End(P_n), lending credence to recent conjectures of Gorsky, Oblomkov, Rasmussen, and Shende [GOR12, GORS12, OS12, ORS12].

PHD (Doctor of Philosophy)
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