An algebraic characterization of a Dehn twist for nonorientable surfaces

TL;DR

This paper provides an algebraic method to identify Dehn twists about simple closed curves on nonorientable surfaces, clarifying previous proofs and extending characterizations to separating curves.

Contribution

It offers a new algebraic characterization of Dehn twists on nonorientable surfaces, including those about separating curves, and addresses gaps in earlier proofs.

Findings

Algebraic characterization of Dehn twists on nonorientable surfaces

Extension of characterizations to separating simple closed curves

Clarification and completion of previous proofs

Abstract

Let $N_g^k$ be a nonorientable surface of genus \ $g\geq 5$ \ with \ $k$-punctures. In this note, we will give an algebraic characterization of a Dehn twist about a simple closed curve on $N_g^k$. Along the way, we will fill some little gaps in the proofs of some theorems in \cite{A} and \cite{I1} giving algebraic characterizations of Dehn twists about separating simple closed curves. Indeed, our results will give an algebraic characterization for the topological type of Dehn twists about separating simple closed curves.