String-Math Seminar: \(gl_n\) homologies, annular evaluation, and symmetric webs

Seminar | October 8 | 2-3 p.m. | 402 LeConte Hall

 David Rose, University of North Carolina at Chapel Hill

 Department of Mathematics

A conjecture of Dunfield-Gukov-Rasmussen predicts a family of differentials on reduced HOMFLY-PT homology, indexed by the integers, that give rise to a corresponding family of reduced link homologies. We'll discuss a variant of this conjecture, constructing an unreduced link homology theory categorifying the quantum \(gl_n\) link invariant for all non-zero values of \(n\) (including negative values!). To do so, we employ the technique of annular evaluation, which uses categorical traces to define and characterize type A link homology theories in terms of simple data assigned to the unknot. Of particular interest is the case of negative n, which gives a categorification of the "symmetric webs" presentation of the type A Reshetikhin-Turaev invariant, and which produces novel categorifications thereof (i.e. distinct from the Khovanov-Rozansky theory).