On powers of half-twists in M(0,2n)

TL;DR

This paper employs skein theory to analyze the subgroup structure generated by powers of half-twists in the mapping class group of a punctured sphere, extending previous results on their index properties.

Contribution

It provides a skein-theoretic proof of Stylianakis's result regarding the infinite index of certain subgroups generated by powers of half-twists.

Findings

Normal closure of m-th powers of half-twists has infinite index under certain conditions

Skein theory effectively analyzes subgroup structures in mapping class groups

Extends previous results on the algebraic properties of half-twists

Abstract

We use elementary skein theory to prove a version of a result of Stylianakis who showed that under mild restrictions on m and n, the normal closure of the m-th power of a half-twist has infinite index in the mapping class group of a sphere with 2n punctures.