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On Heegaard Floer minimal knots in sutured manifolds

Fraser Binns

TL;DR

This work extends the Li–Xie–Zhang classification of minimal knots to the Heegaard Floer setting on balanced sutured manifolds, establishing a rank-based criterion that forces a connect-sum decomposition and Floer simplicity in the closed summand. The authors leverage sutured Floer homology, a Künneth-type formula, and sutured hierarchies to prove that equality $\mathrm{rank}(\mathrm{SFH}(Y(K),\gamma(K)))=2\mathrm{rank}(\mathrm{SFH}(Y,\gamma))$ with $[K]=0$ yields a very rigid knot structure. The results provide further evidence for Kronheimer–Mrowka's conjecture linking instanton and Heegaard Floer homology in applicable cases. In addition, the paper demonstrates that link Floer homology detects spherical braid closures among homologically non-trivial links in $S^1\times S^2$, tying braid theory to HFK rank and enriching the botany of link invariants in this 3-manifold.

Abstract

Li-Xie-Zhang classified instanton Floer minimal knots in balanced sutured manifolds subject to a condition on the fundamental group. In this paper, we give a similar classification in the Heegaard Floer homology setting. Since our classifications agree when they are both applicable, this provides further evidence for the conjecture of Kronheimer-Mrowka that instanton Floer homology and Heegaard Floer homology are isomorphic. We also study link Floer homology botany question in $S^1\times S^2$, showing that link Floer homology detects spherical braid closures among homologically nontrivial links.

On Heegaard Floer minimal knots in sutured manifolds

TL;DR

This work extends the Li–Xie–Zhang classification of minimal knots to the Heegaard Floer setting on balanced sutured manifolds, establishing a rank-based criterion that forces a connect-sum decomposition and Floer simplicity in the closed summand. The authors leverage sutured Floer homology, a Künneth-type formula, and sutured hierarchies to prove that equality with yields a very rigid knot structure. The results provide further evidence for Kronheimer–Mrowka's conjecture linking instanton and Heegaard Floer homology in applicable cases. In addition, the paper demonstrates that link Floer homology detects spherical braid closures among homologically non-trivial links in , tying braid theory to HFK rank and enriching the botany of link invariants in this 3-manifold.

Abstract

Li-Xie-Zhang classified instanton Floer minimal knots in balanced sutured manifolds subject to a condition on the fundamental group. In this paper, we give a similar classification in the Heegaard Floer homology setting. Since our classifications agree when they are both applicable, this provides further evidence for the conjecture of Kronheimer-Mrowka that instanton Floer homology and Heegaard Floer homology are isomorphic. We also study link Floer homology botany question in , showing that link Floer homology detects spherical braid closures among homologically nontrivial links.

Paper Structure

This paper contains 6 sections, 21 theorems, 24 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Sutured Manifolds and Floer homology
  3. Sutured Manifolds
  4. Sutured Floer homology
  5. Floer minimal knots in sutured manifolds
  6. The link Floer homology of homologically non-trivial links in $S^1\times S^2$

Key Result

Theorem 1.1

Suppose that $K$ is a knot in a balanced sutured manifold $(Y,\gamma)$ with $[K]=0\in H_1(Y;{\mathbb{Q}})/H_1(\partial Y;{\mathbb{Q}})$. If $\mathop{\mathrm{rank}}\nolimits(\mathop{\mathrm{SFH}}\nolimits(Y(K),\gamma(K)))=2\mathop{\mathrm{rank}}\nolimits(\mathop{\mathrm{SFH}}\nolimits(Y,\gamma))\neq

Figures (3)

  • Figure 1: The closure of a spherical braid is obtained from a spherical braid $\beta\subset S^2\times[-1,1]$ by identifying $S^2\times\{\pm 1\}$. In the figure, the outer and inner circles indicate $S^2\times\{\pm 1\}$.
  • Figure 2: The torsion relative $\text{spin}^c$-structure on $(D^2\times S^1,\mu)$ can be constructed from the vector field $v$, shown in blue, on the annulus. Here the core of $D^2\times S^1$, $K$, is the inner boundary component of the annulus and oriented counterclockwise. The section of $v^\perp$, restricted to the annulus, points perpendicularly out of the page and decreases in magnitude to $0$ as you move from the outer boundary to the inner boundary. The vector field $v$ on the solid torus can be recovered by rotating the figure about the inner boundary component and modifying the resulting vector field in a neighborhood of the right hand side of the inner boundary component. The section of $v^\perp$ can be defined similarly.
  • Figure 3: $0$-surgery on the meridian --- shown in green --- of the band $\alpha$ --- shown in grey produces a knot in a $3$-manifolds with an $S^1\times S^2$ factor.

Theorems & Definitions (44)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Corollary 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4: gabai1983foliations
  • Definition 2.5
  • ...and 34 more