Instanton Floer homology, sutures, and Heegaard diagrams

TL;DR

This paper introduces a new technique to analyze instanton Floer homology, linking it to Heegaard diagrams, and applies it to compute knot homologies and explore their relations with framed instanton Floer homology.

Contribution

It establishes a novel method connecting instanton Floer homology with Heegaard diagrams and computes instanton knot homology for various (1,1)-knots, including all torus knots in S^3.

Findings

Relation between instanton Floer homology and Heegaard diagrams

Computed instanton knot homology for all torus knots in S^3

Proved inequality between framed instanton Floer homology and knot homology

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.