Math Graduate Student Seminar

In 2016, Gukov, Putrov and Vafa conjectured the existence of invariants of 3-manifolds which are q-series with integer coefficients. They are expected to have a categorification in a sense of Khovanov homology. More recently, in 2019, Gukov and Manolescu studied the analogue of GPV invariants for knot complements and their behavior under Dehn surgery. In this talk, I will review what is known about these conjectural invariants, recalling the results by Gukov-Pei-Putrov-Vafa and Gukov-Manolescu, and then I will discuss a generalization of these results to arbitrary gauge group. If time allows, I will also talk about a HOMFLY-PT analogue of F_K, which is a work in progress with Gruen, Gukov, Kucharski and Sulkowski.