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 (Formula presented.) -manifold or a null-homologous knot inside a (Formula presented.) -manifold and the Heegaard diagram of that (Formula presented.) -manifold or knot. We further use this relation to compute the instanton knot homology of some families of (Formula presented.) -knots, including all torus knots in (Formula presented.), which were mostly unknown before. As a second application, we also study the relation between the instanton knot homology (Formula presented.) and the framed instanton Floer homology (Formula presented.). In particular, we prove the inequality (Formula presented.) for all rationally null-homologous knots (Formula presented.) and we constructed a new decomposition of the framed instanton Floer homology of Dehn surgeries along (Formula presented.) that corresponds to the decomposition along torsion spin (Formula presented.) decompositions in monopole and Heegaard Floer theory.