Seminar

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Given the notion of (combinatorial) Khovanov homology for links, how does one extend this definition for tangles?  Khovanov gave two different, but related, answers, depending on which representation theoretic notion of the "Jones polynomial for tangles" one is aiming to categorify.  In either case, to sets of boundary points we assign rings (called arc algebras in one case, or platform algebras in the other), and then to the tangles we assign modules over these rings.  We will describe these constructions and indicate the sense in which they categorify some representation theoretic tangle Jones polynomial.  We will also discuss some of their basic properties and their Hochschild homologies.  If time permits, we will also discuss the stable homotopy lifts of these invariants due to Lawson-Lipshitz-Sarkar, along with some recent work indicating what aspects of the Hochschild homologies lift to the topological Hochschild homologies (and what does not).