Tuesday, November 17, 2020 10:00 AM
Zhenkun Li (Stanford University)

Instanton knot homology was first introduced by Floer around 1990 and was revisited by Kronheimer and Mrowka around 2010. It is built based on the solution to a set of partial differential equations and is very difficult to compute. On the other hand, Heegaard diagrams are classical tools to describe knots and 3-manifolds combinatorially, and is also the basis of Heegaard Floer theory, which was introduced by Ozsváth and Szabó around 2004. In this talk, I will explain how to extract some information about instanton theory from Heegaard diagrams. In particular, we study the (1,1)-knots, which are known to have simple Heegaard diagrams. We provide an upper bound for the dimension of instanton knot homology for all (1,1)-knots. Also, we prove that, for some families of (1,1)-knots, including all torus knots, the upper bound we obtained is in fact sharp. If time permits, I will also discuss on some further applications to the instanton Floer homology of 3-manifolds coming from Dehn surgeries along null-homologous knots. This is a joint work with Fan Ye.