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An enhanced Euler characteristic of sutured instanton homology

Zhenkun Li, Fan Ye

Abstract

For a balanced sutured manifold $(M,γ)$, we construct a decomposition of $SHI(M,γ)$ with respect to torsions in $H=H_1(M;\mathbb{Z})$, which generalizes the decomposition of $I^\sharp(Y)$ in previous work of the authors. This decomposition can be regarded as a candidate for the counterpart of the torsion spin$^c$ decompositions in $SFH(M,γ)$. Based on this decomposition, we define an enhanced Euler characteristic $χ_{\rm en}(SHI(M,γ))\in\mathbb{Z}[H]/\pm H$ and prove that $χ_{\rm en}(SHI(M,γ))=χ(SFH(M,γ))$. This provides a better lower bound on $\dim_\mathbb{C}SHI(M,γ)$ than the graded Euler characteristic $χ_{\rm gr}(SHI(M,γ))$. As applications, we prove instanton knot homology detects the unknot in any instanton L-space and show that the conjecture $KHI(Y,K)\cong \widehat{HFK}(Y,K)$ holds for all $(1,1)$-L-space knots and constrained knots in lens spaces, which include all torus knots and many hyperbolic knots in lens spaces.

An enhanced Euler characteristic of sutured instanton homology

Abstract

For a balanced sutured manifold , we construct a decomposition of with respect to torsions in , which generalizes the decomposition of in previous work of the authors. This decomposition can be regarded as a candidate for the counterpart of the torsion spin decompositions in . Based on this decomposition, we define an enhanced Euler characteristic and prove that . This provides a better lower bound on than the graded Euler characteristic . As applications, we prove instanton knot homology detects the unknot in any instanton L-space and show that the conjecture holds for all -L-space knots and constrained knots in lens spaces, which include all torus knots and many hyperbolic knots in lens spaces.

Paper Structure

This paper contains 20 sections, 51 theorems, 200 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Organization and sketch of the proofs
  3. Conventions
  4. Acknowledgement
  5. Twisted sutured homology
  6. Notations, gradings, and Euler characteristics
  7. Contact handles and bypasses
  8. Decomposition associated to tangles
  9. Basic setups
  10. One tangle component
  11. More tangle components
  12. Sutured Heegaard Floer homology
  13. Construction and gradings
  14. Euler characteristics
  15. Surgery exact triangle
  16. ...and 5 more sections

Key Result

Theorem 1.1

Suppose $(M,\gamma)$ is a balanced sutured manifold and $H=H_1(M;\mathbb{Z})$. Then there is a (possibly noncaonical) decomposition This decomposition depends on some auxiliary choices. In particular, it is defined up to a global shift of $H$. We define the enhanced Euler characteristic of $SHI$ by Then we have The similar results also hold for $SHM(M,\gamma)$.

Figures (12)

  • Figure 1: The positive and negative stabilizations of $S$.
  • Figure 2: Left, the sutured manifold $(M,\gamma)$ with two points $p$ and $q$ on the suture. Right, the 1-handle attachment along $p$ and $q$.
  • Figure 3: Left, the sutured manifold $(M,\gamma)$ and the curve $\beta\subset\partial M$ that intersects $\gamma$ at two points. Right, the 2-handle attachment along the curve $\mu$.
  • Figure 4: The bypass arc and the bypass attachment, where the orientation of $\partial M$ is pointing out.
  • Figure 5: The bypass triangle.
  • ...and 7 more figures

Theorems & Definitions (94)

  • Theorem 1.1: Main theorem
  • Remark 1.2
  • Theorem 1.3
  • Example 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 84 more