Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonexistence and existence of fillable contact structures on 3-manifolds

Fan Ding, Youlin Li, Zhongtao Wu

TL;DR

The paper investigates when Dehn surgeries on L-space knots and links yield 3-manifolds that admit (non)Stein fillable contact structures. It builds nonexistence results for certain surgery ranges using Heegaard Floer $d$-invariants and related obstructions, while also proving existence results for Stein fillable structures at sufficiently large surgery coefficients, and introducing the Stein fillable coefficient $Sfc(K)$ to quantify when Stein fillings arise. The work covers explicit families, including twisted torus knots $K_{n,m}$, $K'_{n,m}$, two-component L-space links like $\mathbb{L}_{n}$, and two-bridge links, deriving explicit thresholds and conditions. It also formulates and motivates broader questions, such as the high-surgery conjecture and the potential behavior of Stein fillability for links, and provides precise calculations and constructions (e.g., Stein handlebody realizations) to support these claims. Overall, the paper advances understanding of how surgery parameters interact with contact-geometry fillability, offering concrete families with both nonexistence and Stein-fillability results and introducing invariants to guide future investigations.

Abstract

In the first part of this paper, we construct infinitely many hyperbolic closed 3-manifolds which admit no symplectic fillable contact structure. All these 3-manifolds are obtained by Dehn surgeries along L-space knots or L-space two-component links. In the second part of this paper, we show that Dehn surgeries along certain knots and links, including those considered in the first part, admit Stein fillable contact structures as long as the surgery coefficients are sufficiently large. This provides some new evidence for the high surgery conjecture raised by Stipsicz.

Nonexistence and existence of fillable contact structures on 3-manifolds

TL;DR

The paper investigates when Dehn surgeries on L-space knots and links yield 3-manifolds that admit (non)Stein fillable contact structures. It builds nonexistence results for certain surgery ranges using Heegaard Floer -invariants and related obstructions, while also proving existence results for Stein fillable structures at sufficiently large surgery coefficients, and introducing the Stein fillable coefficient to quantify when Stein fillings arise. The work covers explicit families, including twisted torus knots , , two-component L-space links like , and two-bridge links, deriving explicit thresholds and conditions. It also formulates and motivates broader questions, such as the high-surgery conjecture and the potential behavior of Stein fillability for links, and provides precise calculations and constructions (e.g., Stein handlebody realizations) to support these claims. Overall, the paper advances understanding of how surgery parameters interact with contact-geometry fillability, offering concrete families with both nonexistence and Stein-fillability results and introducing invariants to guide future investigations.

Abstract

In the first part of this paper, we construct infinitely many hyperbolic closed 3-manifolds which admit no symplectic fillable contact structure. All these 3-manifolds are obtained by Dehn surgeries along L-space knots or L-space two-component links. In the second part of this paper, we show that Dehn surgeries along certain knots and links, including those considered in the first part, admit Stein fillable contact structures as long as the surgery coefficients are sufficiently large. This provides some new evidence for the high surgery conjecture raised by Stipsicz.

Paper Structure

This paper contains 10 sections, 32 theorems, 92 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Non-existence of symplectic fillable contact structures
  4. Dehn surgeries along L-space knots
  5. Dehn surgeries along two-component L-space links
  6. Non-L-space
  7. Existence of Stein fillable contact structures
  8. Sufficiently large Dehn surgeries along knots
  9. Stein fillable coefficients
  10. Sufficiently large Dehn surgeries along links

Key Result

Theorem 1.2

Suppose $K$ is a closed 3-braid which is an L-space knot, then the rational $r$-surgery along $K$ yields a 3-manifold admitting no symplectic fillable contact structure for $r\in[2g(K)-1, 2g(K)]$, where $g(K)$ is the genus of $K$. Precisely speaking, the 3-manifold $S^{3}_{r}(K_{n,m})$ (resp. $S^{3}

Figures (13)

  • Figure 1: The left picture is the twisted torus knot $K_{n,m}$. The right picture is the twisted torus knot $K'_{n,m}$. The boxes represent $m$ full right handed twists and $n-2$ full right handed twists.
  • Figure 2: Two-component link $\mathbb{L}_n$. The box represents $n$ full right handed twists.
  • Figure 3: Two-bridge link $K(a_1,a_2)$. For $i=1,2$, if $a_i\geq0$, then there are $a_i$ positive crossings. If $a_i<0$, then there are $-a_i$ negative crossings. The link $K(a_1,a_2)$ is a knot if and only if either $a_1$ or $a_2$ is even.
  • Figure 4: Surgery on twisted torus knot $T(u,um+1;u-1,l)$. Both of the two braces indicate $u-4$ two-strand braids each of which consists of a left-handed full twist.
  • Figure 5: Surgery on twisted torus knot $T(u,um+u-1;u-1,l)$. Both of the two braces indicate $u-2$ two-strand braids each of which consists of a left-handed full twist. In the diagrams (3)-(7), the knots without labels all have surgery coefficients $-\frac{1}{l+1}$.
  • ...and 8 more figures

Theorems & Definitions (69)

  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Proposition 1.7
  • Remark 1.8
  • Proposition 1.11
  • Theorem 1.13
  • Theorem 1.14
  • ...and 59 more