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Dmitry Vagner, Duke University, “A smooth TQFT approach to sln homology”

April 19, 2017 | 4:00 pm - 5:00 pm EDT

Given a link diagram L, one can apply a Skein relation to each crossing to yield a cube of resolutions. These skein relations come from the braiding in the category of Uq(sln) representations. When n2, we have the Khovanov cube of resolutions with edge maps defined by (co)pants conordisms. We may then apply a smooth TQFT, flatten the complex, and take homology to extract Khovanov homology. The case is more complicated when n2 because, due to the higher complexity of morphisms among fundamental representations, the resolution diagrams will contain trivalent vertices. The first definition, due to Khovanov and Rozansky, defined the invariants in the category of matrix factorizations. A more diagrammatic and combinatorial approach, taken by Queffelef-Rose, considers foam cobordisms—those with seams—between trivalent graphs and exploits skew-Howe duality to define a foam-like TQFT on them so as to extract homology. In this work, joint with Michael Abel, we apply a virtual filtration—another Skein-like relation, justified by the behavior of matrix factorizations—to resolve all trivalent graphs to their own cubes of smooth resolutions. With a little work, this yields a triple complex from which we extract a spectral sequence whose second page is equivalent to the Khovanov-Rozansky sln homology of the link.

Details

Date:
April 19, 2017
Time:
4:00 pm - 5:00 pm EDT
Event Category:

Venue

SAS 4201