Abstract: Since Floer's work in 1988, various Floer homologies have been constructed for closed 3-manifolds, knots, and sutured manifolds. In 2008, Kronheimer-Mrowka proposed a conjecture about isomorphisms among Floer homologies. The isomorphisms among Heegaard Floer, Seiberg-Witten, and embedded contact homology are established by many groups of people, while the relation to instanton Floer homology remains open.
In this talk, I will first introduce the history of the Floer theory and then introduce a dimension inequality between Heegaard Floer knot homology and instanton knot homology. The proof is based on the combinatorial version of Heegaard Floer homology, originally introduced by Sarkar-Wang. This work is joint with Baldwin, Li, and Sivek.
