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 $I^{♯}(Y)$. In particular, we prove the inequality ${\dim }_{\mathbb{C}}{I}^{♯}(Y)\le {\dim }_{\mathbb{C}}KHI(Y,K)$ 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.

61 pages, 24 figures; accepted by Journal of Topology; v3: we removed some results in $\hat{A}\S$ 4.4-4.7 of the previous version to make the paper shorter