Open book decompositions with page a four-punctured sphere

TL;DR

This work analyzes contact structures supported by open books with page $\Sigma_{0,4}$. It combines open book foliations with bordered Floer techniques to classify when reducible monodromies yield tight or overtwisted structures and to determine which reducible monodromies have nonzero Heegaard Floer contact invariants. The first part constructs infinite families of overtwisted, right-veering monodromies and gives a precise tightness criterion for reducible cases; the second part reestablishes (via bordered invariants) when reducible monodromies have nonzero invariants, aligning Stein fillability with non-vanishing Floer data. Together, these results deepen understanding of the Giroux correspondence on four-punctured-sphere pages and showcase the power of bordered Floer methods in tracking contact invariants through monodromy data.

Abstract

In this paper, we study contact structures supported by open book decompositions whose pages are four-punctured spheres. The paper is split into two parts. In the first part, we find infinitely many overtwisted, right-veering monodromies on the four-punctured sphere. This is done using the techniques developed by Ito-Kawamuro in the papers arXiv:1112.5874, arXiv:1310.6404. Although most of the monodromies that we show are overtwisted are pseudo-Anosov, we are also able to classify precisely which reducible monodromies on the four-punctured sphere are tight. In the second part of the paper, we reprove part of a result of Lekili arXiv:1008.3529 by classifying which reducible mondromies have non-zero Heegaard Floer invariant. This is done by using the bordered contact invariants of Min-Varvarezos arXiv:2410.05511.

Figures (82)

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Theorems & Definitions (58)

• Theorem 1.1: Ito--Kawamuro
• Theorem 1.2
• Theorem 1.3: Lekili
• Proposition 3.1
• proof
• Remark 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof