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Connected sums and directed systems in knot Floer homologies

Sudipta Ghosh, Ian Zemke

Abstract

We prove a number of fundamental properties about instanton knot Floer homology. Our arguments rely on general properties of sutured Floer theories and apply also in the Heegaard Floer and monopole Floer settings, where many of our results were already known. Our main result is the connected sum formula for instanton knot Floer homology. An extension of this result proves the oriented skein exact triangle for the minus version of instanton knot Floer homology. Finally, we derive a new model of the minus version of instanton knot Floer homology, which takes the form of a free, finitely generated chain complex over a polynomial ring, as opposed to a direct limit. This construction is new to all of the Floer theories. We explore these results also in the context of Heegaard Floer theory as well.

Connected sums and directed systems in knot Floer homologies

Abstract

We prove a number of fundamental properties about instanton knot Floer homology. Our arguments rely on general properties of sutured Floer theories and apply also in the Heegaard Floer and monopole Floer settings, where many of our results were already known. Our main result is the connected sum formula for instanton knot Floer homology. An extension of this result proves the oriented skein exact triangle for the minus version of instanton knot Floer homology. Finally, we derive a new model of the minus version of instanton knot Floer homology, which takes the form of a free, finitely generated chain complex over a polynomial ring, as opposed to a direct limit. This construction is new to all of the Floer theories. We explore these results also in the context of Heegaard Floer theory as well.
Paper Structure (54 sections, 52 theorems, 178 equations, 24 figures)

This paper contains 54 sections, 52 theorems, 178 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Results
  3. A limit free definition of $\mathit{KHI}^-(Y,K)$
  4. Organization
  5. Acknowledgments
  6. Background on sutured Floer theories
  7. Sutured Heegaard Floer homology
  8. Sutured instanton Floer homology
  9. Sutured monopole Floer homology
  10. Contact handle maps in Floer theories
  11. Contact 1-handles
  12. Contact 2-handles
  13. On the Heegaard Floer contact 2-handle map
  14. Background on directed systems in knot Floer homologies
  15. Bypass attachment
  16. ...and 39 more sections

Key Result

Theorem 1.1

Suppose that $K_1,K_2$ are null-homologous knots in 3-manifolds $Y_1$ and $Y_2$, respectively. Then the instanton limit Floer homology $\mathit{KHI}^-(Y_1\# Y_2, K_1\#K_2)$ is isomorphic to the derived tensor product as an $\mathbb{C}[U]$-module.

Figures (24)

  • Figure 2.1: Left: the sutured manifold $(M, \gamma)$ with two points $P$ and $Q$ on the suture. Right: the manifold $(M',\gamma')$ obtained by attaching a contact 1-handle with feet at $P$ and $Q$.
  • Figure 2.2: The diagrams used in the definition of the contact handle maps
  • Figure 2.3: Left: the sutured manifold $(M, \gamma)$ and the curve $\mu \subseteq \partial M$ that intersects $\gamma$ at two points. Right: the contact 2-handle attachment along the curve $\mu$. In both pictures, we view $M$ and $M'$ as being the exterior of the surfaces shown.
  • Figure 2.4: The Heegaard diagrams used in the definition of the contact 2-handle map. On the left is $(\Sigma,\boldsymbol{\mathbf{\alpha}},\boldsymbol{\mathbf{\beta}})$ and on the right is $(\Sigma',\boldsymbol{\mathbf{\alpha}}\cup \{\alpha_0\}, \boldsymbol{\mathbf{\beta}}\cup \{\beta_0\})$.
  • Figure 2.5: Realizing the contact 2-handle map $C_{h^2}$ as a composition of a compound stabilization $\sigma$, followed by a 4-dimensional 2-handle map $F_{W}$, followed by the inverse of the contact 1-handle map $(C_{h^1})^{-1}$. A holomorphic triangle of the 2-handle map is indicated in the right-most box.
  • ...and 19 more figures

Theorems & Definitions (107)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3: KMSutures*Corrollary 7.2
  • Definition 2.4: KMskein*Definition 7.3
  • Definition 2.5: KMSutures*Section 7.4
  • Definition 2.6: KMSutures*Definition 7.10
  • ...and 97 more