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 $\hskip.001pt 3$ -manifold or a null-homologous knot inside a 3 $\hskip.001pt 3$ -manifold and the Heegaard diagram of that 3 $\hskip.001pt 3$ -manifold or knot. We further use this relation to compute the instanton knot homology of some families of ( 1 , 1 ) $(1,1)$ -knots, including all torus knots in S 3 $S^3$ , which were mostly unknown before. As a second application, we also study the relation between the instanton knot homology K H I ( Y , K ) $KHI(Y,K)$ and the framed instanton Floer homology I ♯ ( Y ) $I^\sharp (Y)$ . In particular, we prove the inequality dim C I ♯ ( Y ) ⩽ dim C K H I ( Y , K ) $\dim _\mathbb {C} I^\sharp (Y)\leqslant \dim _\mathbb {C}KHI(Y,K)$ for all rationally null-homologous knots K ⊂ Y $K\subset Y$ and we constructed a new decomposition of the framed instanton Floer homology of Dehn surgeries along K $K$ that corresponds to the decomposition along torsion spin c ${}^c$ decompositions in monopole and Heegaard Floer theory.