Positive factorizations of symmetric mapping classes

TL;DR

This paper investigates whether symmetric mapping classes with positive factorizations are lifts of quasi-positive braids, providing an affirmative answer under specific cyclic conditions.

Contribution

It offers a new criterion linking positive factorizations of symmetric mapping classes to their being lifts of quasi-positive braids under certain cyclic conditions.

Findings

Affirmative answer for symmetric mapping classes with certain cyclic properties

Establishes a connection between positive factorizations and lifts of quasi-positive braids

Provides conditions under which the conjecture holds

Abstract

We study Question 7.9 in the paper "Monoids in the mapping class group" by Etnyre and Van Horn-Morris; whether a symmetric mapping class admitting a positive factorization is a lift of a quasi-positive braid. We answer affirmatively for mapping classes satisfying certain cyclic conditions.