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Tight contact structures on toroidal plumbed 3-manifolds

Tanushree Shah, Jonathan Simone

TL;DR

The paper investigates tight contact structures on plumbed 3-manifolds with no bad vertices, focusing on those with zero Giroux torsion and examining when torsion can be added without creating overtwisted structures. It introduces rigidity and maximal chains, develops an algorithm to construct Stein diagrams for trivalent plumbings, and provides both lower and upper bounds on the number of tight structures, including a constructive count of Stein fillable cases. In the trivalent case, the authors establish a lower bound of $$(a_1-1)\cdots skip(a_n-1)$$ Stein fillable tight structures and outline an upper bound for a specific family, using convex surface theory to analyze twisting. The work also discusses higher-valence plumbings, proposes a conjecture about universal tightness of toric annuli in the presence of torsion, and outlines extensions of the methods to more general graphs, offering a practical framework for counting and realizing tight, torsion-controlled contact structures on plumbed 3-manifolds with potential implications for Stein fillings.

Abstract

We consider tight contact structures on plumbed 3-manifolds with no bad vertices. We discuss how one can count the number of tight contact structures with zero Giroux torsion on such 3-manifolds and explore conditions under which Giroux torsion can be added to these tight contact structures without making them overtwisted. We give an explicit algorithm to construct stein diagrams corresponding to tight structures without Giroux torsion. We focus mainly on plumbed 3-manifolds whose vertices have valence at most 3 and then briefly consider the situation for plumbed 3-manifolds with vertices of higher valence.

Tight contact structures on toroidal plumbed 3-manifolds

TL;DR

The paper investigates tight contact structures on plumbed 3-manifolds with no bad vertices, focusing on those with zero Giroux torsion and examining when torsion can be added without creating overtwisted structures. It introduces rigidity and maximal chains, develops an algorithm to construct Stein diagrams for trivalent plumbings, and provides both lower and upper bounds on the number of tight structures, including a constructive count of Stein fillable cases. In the trivalent case, the authors establish a lower bound of Stein fillable tight structures and outline an upper bound for a specific family, using convex surface theory to analyze twisting. The work also discusses higher-valence plumbings, proposes a conjecture about universal tightness of toric annuli in the presence of torsion, and outlines extensions of the methods to more general graphs, offering a practical framework for counting and realizing tight, torsion-controlled contact structures on plumbed 3-manifolds with potential implications for Stein fillings.

Abstract

We consider tight contact structures on plumbed 3-manifolds with no bad vertices. We discuss how one can count the number of tight contact structures with zero Giroux torsion on such 3-manifolds and explore conditions under which Giroux torsion can be added to these tight contact structures without making them overtwisted. We give an explicit algorithm to construct stein diagrams corresponding to tight structures without Giroux torsion. We focus mainly on plumbed 3-manifolds whose vertices have valence at most 3 and then briefly consider the situation for plumbed 3-manifolds with vertices of higher valence.

Paper Structure

This paper contains 13 sections, 13 theorems, 9 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Higher Valence
  3. A Conjecture
  4. Organization
  5. Acknowledgements
  6. Rigidity
  7. Maximal Chains
  8. An Example
  9. Stein Diagrams: Proof of Theorem \ref{['thm:mainminimallytwisting']}
  10. $\Gamma$ is planar and 2-connected
  11. $\Gamma$ is nonplanar
  12. General Case: Proof of Theorem \ref{['thm:mainminimallytwisting']}
  13. Plumbed 3-manifolds with higher valence vertices

Key Result

Theorem 2

Let $M$ be a plumbed 3-manifold with no bad vertices and whose vertices are all at most trivalent. Let the weights of the vertices be $-a_1,\ldots,-a_n$. Then $M$ admits at least $(a_1-1)\cdots(a_n-1)$ Stein fillable contact structures.

Figures (13)

  • Figure 1: A plumbed 3-manifold
  • Figure 2: Plumbed 3-manifold $Y$
  • Figure 3: Maximal Chain in $Y$
  • Figure 4: Recutting $Y$
  • Figure 5: Plumbed 3-manifold $Y$
  • ...and 8 more figures

Theorems & Definitions (29)

  • Theorem 2
  • Remark 3
  • Theorem 4
  • Remark 5
  • Remark 6
  • Conjecture 7
  • Remark 8
  • Lemma 9
  • proof
  • Proposition 10: c.f. EMM
  • ...and 19 more