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$3$$3$-manifold or a null-homologous knot inside a 3$3$$3$-manifold and the Heegaard diagram of that 3$3$$3$-manifold or knot. We further use this relation to compute the instanton knot homology of some families of (1,1)$(1,1)$$(1,1)$-knots, including all torus knots in 𝑆3$S^3$$S^3$, which were mostly unknown before. As a second application, we also study the relation between the instanton knot homology 𝐾𝐻𝐼(𝑌,𝐾)$KHI(Y,K)$$KHI(Y,K)$ and the framed instanton Floer homology 𝐼♯(𝑌)$I^{♯}(Y)$$I^{♯}(Y)$. In particular, we prove the inequality dimℂ𝐼♯(𝑌)⩽dimℂ𝐾𝐻𝐼(𝑌,𝐾)${dim}_{\mathbb{C}}{I}^{♯}(Y)⩽{dim}_{\mathbb{C}}KHI(Y,K)$${dim}_{\mathbb{C}}{I}^{♯}(Y)⩽{dim}_{\mathbb{C}}KHI(Y,K)$ for all rationally null-homologous knots 𝐾⊂𝑌$K\subset Y$$K\subset Y$ and we constructed a new decomposition of the framed instanton Floer homology of Dehn surgeries along 𝐾$K$$K$ that corresponds to the decomposition along torsion spin𝑐${}^c$${}^c$ decompositions in monopole and Heegaard Floer theory.

中文翻译：

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Instanton Floer 同源性、缝合线和 Heegaard 图

本文建立了一种新技术，使我们能够访问瞬时 Floer 同源性的一些基本结构特性。作为一个应用程序，我们首次建立了一个瞬时子 Floer 同调之间的关系3$3$$3$-manifold 或 a 内的空同源结3$3$$3$-manifold 和它的 Heegaard 图3$3$$3$-歧管或结。我们进一步使用这种关系来计算一些家族的瞬时节点同源性( 1 , 1 )$(1,1)$$(1,1)$-结，包括所有圆环结𝑆3${\mathrm{小号}}^3$$S^3$，这在以前大多是未知的。作为第二个应用，我们还研究了瞬时结同源性之间的关系𝐾𝐻𝐼( 𝑌, 𝐾)$ķH\mathrm{一世}(是,ķ)$$KHI(Y,K)$和成帧的瞬时子 Floer 同调𝐼♯( 𝑌)${\mathrm{一世}}^{♯}(是)$$I^{♯}(Y)$. 特别地，我们证明了不等式暗淡ℂ𝐼♯( 𝑌) ⩽暗淡ℂ𝐾𝐻𝐼( 𝑌, 𝐾)${暗淡}_{\mathbb{C}}{\mathrm{一世}}^{♯}(是)⩽{暗淡}_{\mathbb{C}}ķH\mathrm{一世}(是,ķ)$${dim}_{\mathbb{C}}{I}^{♯}(Y)⩽{dim}_{\mathbb{C}}KHI(Y,K)$对于所有理性零同源结𝐾⊂𝑌 _$ķ\subset 是$$K\subset Y$我们构建了一个新的 Dehn 手术框架瞬时 Floer 同源性分解𝐾$ķ$$K$对应于沿扭转自旋的分解𝑐${}^C$${}^c$单极子和 Heegaard Floer 理论中的分解。