GT seminar: Tangle invariants valued in derived category of sheaves

Abstract: In this talk, I report on the joint project with Alex Takeda. In their work, Cautis and Kamnitzer constructed a family of varieties Y_n, and associate to a tangle of m incoming and n outgoing strands a functor between derived category of categories of sheaves on Y_m and Y_n. More specifically, this functor is a Fourier Mukai transform, so the invariant assigned to the tangle is the FM kernel. This gives an alternative construction of categorification of the Jones polynomial.

Our main result is to introduce the tilting complex into the computation, so that the construction is lifted up to the homotopy category, and the invariant assigned to a tangle is an explicit complex of projectives in the homotopy category of modules of the quiver algebra associated to Y_n.

This is a talk in our Geometry and Topology Seminar Series.