Simple expressions for the holed torus relations

TL;DR

This paper simplifies existing factorizations of boundary multi-twists in the mapping class group of k-holed tori, providing more concise expressions for these relations across 0 to 9 holes.

Contribution

It offers simplified, more elegant expressions for the k-holed torus relations, improving upon previous explicit factorizations.

Findings

Simplified expressions for boundary multi-twists in the mapping class group.

Unified approach for all k between 0 and 9.

Enhanced clarity and usability of the relations.

Abstract

In the mapping class group of a $k$-holed torus with $0 \leq k \leq 9$, one can factorize the boundary multi-twist (or the identity when $k=0$) as the product of twelve right-handed Dehn twists. Such factorizations were explicitly given by Korkmaz and Ozbagci for each $k \leq 9$ and an alternative one for $k=8$ by Tanaka. In this note, we simplify their expressions for the $k$-holed torus relations.