Open book decompositions with page a four-punctured sphere
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.