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 nullhomologous 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 ℂ 𝐼 ♯ (𝑌) ⩽ dim ℂ 𝐾𝐻𝐼(𝑌, 𝐾) for all rationally null-homologous knots 𝐾 ⊂ 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.

An integral surgery formula for framed instanton homology I

“…Note that there are Z-gradings (see Section 2.1 for details) on SHIp´S 3 pKq, ´Γ 2m`2k´1 2 q and SHIp´S 3 pKq, ´Γm`2k´1 q as in [Li19, Section 3]. From [LY22, Theorem 1.12] (see [LY22,Proposition 4.26] for details), we have…”

“…Proposition 1.16), which might be of independent interest. The commutative diagram (1.12) is entirely based on the commutativity of bypass maps that the authors already proved in [LY22,Section 4.4]. The last commutative diagram (1.13) is essentially a new one.…”

Sutured instanton homology and Heegaard diagrams

“…for sufficiently large m. However, in fact, this inequality can be viewed as coming from a spectral sequence similar to that in [LY22,§ 4]. This spectral sequence can also be described as follows.…”

“…The endpoints of the arc agree with , and . We can, thus, use together with and to define a sequence of sutures for , as in [LY22, § 3.2], which one should regard as ‘longitudinal’ sutures for ; see Figure 5. By the construction of and the assumption that is rationally nullhomologous in , we have Therefore, by [LY22, Lemmas 3.21 and 3.22], we have the following.…”

“…A grading by homotopy classes of 2-plane fields on would enable one to define decompositions of these groups, as well as an analogue of the Maslov grading in Heegaard Floer homology (see [LY22, § 4] for another approach to such a decomposition). The current lack of such structure makes it difficult to translate arguments from the Heegaard Floer setting to the instanton Floer setting.…”