[Paper Review] Overtwisted discs in planar open books

Tetsuya Ito, Keiko Kawamuro|arXiv (Cornell University)|Jan 14, 2014

Geometric and Algebraic Topology15 references3 citations

TL;DR

This paper introduces a topological method using open book foliations to show that overtwisted discs in planar open books can be isotoped into a 'nice' position where all valence ≤1 vertices in the negative graph are strongly essential. As a key result, it proves that if all fractional Dehn twist coefficients (FDTCs) of a planar open book exceed 1, the supported contact structure is tight—offering a new, purely topological criterion for tightness without requiring analysis of Reeb dynamics or Nielsen-Thurston types.

ABSTRACT

Using open book foliations we show that an overtwisted disc in a planar open book can be put in a topologically nice position. As a corollary, we prove that a planar open book whose fractional Dehn twist coefficients grater than one for all the boundary components supports a tight contact structure.

Motivation & Objective

• To establish a topological criterion for tightness of contact structures supported by planar open books.
• To show that overtwisted discs in such open books can be isotoped into a canonical, topologically well-behaved position.
• To prove that if all fractional Dehn twist coefficients exceed 1, the contact structure is tight, even without assuming monodromy type or fillability.
• To provide a combinatorial, topological alternative to analytic methods like contact homology for detecting tightness.

Proposed method

• The authors use open book foliations to analyze the intersection of an overtwisted disc with the pages of a planar open book.
• They define a complexity invariant $\mathfrak{C}(D)$ on the disc's foliation graph to guide isotopy moves that simplify the disc's position.
• A key operation—deforming at non-strongly essential valence ≤1 vertices—reduces complexity while preserving transverse isotopy type of the boundary.
• The construction relies on properties of the negative graph $G_{--}(D)$ and nesting levels of vertices to ensure that all valence ≤1 vertices become strongly essential.
• The method avoids analytic tools like Reeb vector fields or contact forms, relying instead on topological and combinatorial data from the open book.
• The proof uses induction on complexity, showing that any overtwisted disc can be isotoped to satisfy condition (SE1), which is then used to derive the main corollary.

Experimental results

Research questions

• RQ1 Can overtwisted discs in planar open books be isotoped into a topologically canonical position with controlled vertex behavior?
• RQ2 Does a planar open book with all fractional Dehn twist coefficients greater than 1 necessarily support a tight contact structure?
• RQ3 Is there a purely topological criterion for tightness that avoids reliance on contact homology or Reeb dynamics?
• RQ4 Can the complexity invariant $\mathfrak{C}(D)$ be used to systematically simplify open book foliations of surfaces in planar open books?
• RQ5 What is the minimal topological condition on FDTCs that guarantees tightness in planar open books?

Key findings

• An overtwisted disc in a planar open book can be isotoped so that all valence ≤1 vertices in the negative graph $G_{--}(D)$ are strongly essential, satisfying condition (SE1).
• If all fractional Dehn twist coefficients $c(\phi, C) > 1$ for every boundary component $C$ of the page surface, then the open book supports a tight contact structure.
• The criterion is sharp: examples exist with $c(\phi, C) = 1$ on some components and overtwisted structures, showing the bound cannot be weakened.
• The method provides a topological alternative to analytic invariants like contact homology, avoiding the need for Reeb vector field analysis.
• The complexity reduction process via vertex deformation preserves transverse isotopy type of the boundary, ensuring the isotopy is well-behaved.
• The result holds even for non-destabilized open books, demonstrating the robustness of the criterion.