论文标题
Instanton Floer同源性，缝合线和Heegaard图
Instanton Floer homology, sutures, and Heegaard diagrams
论文作者
论文摘要
本文建立了一种新技术，使我们能够访问Instanton 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$ 3 $ manifold的Instanton Floer同源性之间的关系，或者在$ 3 $ MANIFOLD内的无效结与那个$ 3 $ manifold或打结的Heegaard图。我们进一步使用此关系来计算一些$（1,1）$ - 结的家庭的instanton打结同源性，包括$ s^3 $中的所有圆环结，这些结中以前大多是未知的。作为第二个应用程序，我们还研究了Instanton打结同源性$ KHI（y，k）$与框架Instanton Floer同源性$ i^\ sharp（y）$之间的关系。 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沿Torsion Spin $^C $分解单调和Heegaard浮子理论。
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.
