Monodromy factorizations of Seifert fibered spaces

TL;DR

This paper investigates relations in the mapping class monoid of punctured spheres, focusing on factorizations related to open book decompositions of contact 3-manifolds, and computes properties of their symplectic fillings.

Contribution

It introduces new relations between boundary parallel and non boundary parallel twists, enabling analysis of different symplectic fillings of contact 3-manifolds.

Findings

Computed Euler characteristics of fillings from different factorizations

Presented plumbing graphs for specific fillings

Provided bounds on Euler characteristics for fillings

Abstract

We present relations in the mapping class monoid of $S_{0,0}^n$ between products of boundary parallel twists and those involving only non boundary parallel twists. These are of particular interest because each element gives an open book decomposition of a contact 3-manifold, and different factorizations of the same mapping class give potentially distinct symplectic fillings with diffeomorphic boundaries. We apply these results in order to compute the Euler characteristics of fillings resulting from different factorizations, present plumbing graphs for the fillings given by products of boundary parallel twists, and provide sharp bounds on the Euler characteristics for fillings arising from such relations.