Instanton Floer homology, sutures, and Heegaard diagrams

Abstract

This paper establishes a new technique that enables us to access some fundamental structural properties of instanton Floer homology. As an application, we establish, for the first time, a relation between the instanton Floer homology of a $3$-manifold or a null-homologous knot inside a $3$-manifold and the Heegaard diagram of that $3$-manifold or knot. We further use this relation to compute the instanton knot homology of some families of $(1,1)$-knots, including all torus knots in $S^3$, which were mostly unknown before. As a second application, we also study the relation between the instanton knot homology $KHI(Y,K)$ and the framed instanton Floer homology $I^\sharp(Y)$. In particular, we prove the inequality $\dim_\mathbb{C} I^\sharp(Y)\le \dim_\mathbb{C}KHI(Y,K)$ for all rationally null-homologous knots $K\subset Y$ and we constructed a new decomposition of the framed instanton Floer homology of Dehn surgeries along $K$ that corresponds to the decomposition along torsion spin$^c$ decompositions in monopole and Heegaard Floer theory.