Fixed points, Khovanov homology, and Dehn surgery
We use a relationship between Heegaard Floer homology and the symplectic Floer homology of surface diffeomorphisms to partially characterize knots with the same knot Floer homology as the torus knot T(2,5). We then combine this with classical results on the dynamics of surface homeomorphisms, and tools from gauge theory, Khovanov homology, and Khovanov homotopy to prove that Khovanov homology detects T(2,5). The ideas introduced in this work have also recently been used to solve problems in Dehn surgery stemming from Kronheimer and Mrowka's resolution of the Property P conjecture, which I will survey if there is time. This is mostly joint work with Ying Hu and Steven Sivek.