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Detecting Heegaard Floer homology solid tori

Akram Alishahi, Tye Lidman, Robert Lipshitz

TL;DR

The paper establishes a sharp geometric criterion for Heegaard Floer homology solid tori (HFSTs): a rational homology solid torus $M$ is an HFST iff the Dehn filling along its rational longitude, $M(\lambda)$, contains a non-separating $S^2$. It leverages both the classical bordered Floer framework and the immersed-curve reformulation to relate HFSTs to twisted surgery data and to show that all HFSTs arise as knot complements in reducible manifolds. It then classifies Seifert-fibered HFSTs, showing they occur precisely when the base orbifold is a Möbius band or when the filled space is $D^2(0;p/q,-p/q)$. The results connect bordered Floer theory, twisted coefficients, and immersed-curve techniques to provide a complete picture of HFSTs and their Seifert-fibered instances, with implications for L-space fillings and non-separating sphere phenomena.

Abstract

We show that a rational homology solid torus is a Heegaard Floer homology solid torus if and only if it has a Dehn filling with a non-separating 2-sphere. Using this, we characterize Seifert fibered Heegaard Floer solid tori.

Detecting Heegaard Floer homology solid tori

TL;DR

The paper establishes a sharp geometric criterion for Heegaard Floer homology solid tori (HFSTs): a rational homology solid torus is an HFST iff the Dehn filling along its rational longitude, , contains a non-separating . It leverages both the classical bordered Floer framework and the immersed-curve reformulation to relate HFSTs to twisted surgery data and to show that all HFSTs arise as knot complements in reducible manifolds. It then classifies Seifert-fibered HFSTs, showing they occur precisely when the base orbifold is a Möbius band or when the filled space is . The results connect bordered Floer theory, twisted coefficients, and immersed-curve techniques to provide a complete picture of HFSTs and their Seifert-fibered instances, with implications for L-space fillings and non-separating sphere phenomena.

Abstract

We show that a rational homology solid torus is a Heegaard Floer homology solid torus if and only if it has a Dehn filling with a non-separating 2-sphere. Using this, we characterize Seifert fibered Heegaard Floer solid tori.
Paper Structure (7 sections, 7 theorems, 7 equations, 2 figures)

This paper contains 7 sections, 7 theorems, 7 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Classical bordered Floer homology
  4. Bordered Floer homology via immersed curves
  5. Knot complements in reducible manifolds
  6. All HFSTs arise as such
  7. Seifert fibered HFSTs

Key Result

Theorem 1.1

Let $M$ be a rational homology solid torus and let $\lambda$ denote the rational longitude. Then $M$ is a Heegaard Floer homology solid torus if and only if $M(\lambda)$ contains a non-separating 2-sphere.

Figures (2)

  • Figure 1: An example of the first case in the proof of Proposition \ref{['prop:other-easy-dir']}. The longitude is dashed and the immersed curve is solid; this pair is admissible.
  • Figure 2: An example of the second case in the proof of Proposition \ref{['prop:other-easy-dir']}. Left: the immersed curve invariant, with conventions as in Figure \ref{['fig:admissible-case']}. This pair is not admissible. Center: the corresponding bordered invariant. Right: the tensor product with $\underline{\mathcal{S}}$.

Theorems & Definitions (14)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • ...and 4 more