Curves on non--orientable surfaces and crosscap transpositions

TL;DR

This paper provides explicit formulas for how crosscap transpositions act on multicurves in non-orientable surfaces, enhancing understanding of their mapping class groups through generalized Dynnikov coordinates.

Contribution

It introduces explicit formulas for the action of crosscap transpositions on multicurves in non-orientable surfaces using generalized Dynnikov coordinates.

Findings

Explicit formulas for crosscap transpositions' action on multicurves.

Enhanced understanding of the mapping class group structure.

Application of generalized Dynnikov coordinates to non-orientable surfaces.

Abstract

Let $N_{g,n}$ be an $n$--punctured non--orientable surface of genus $g$ with one boundary component. For $g\geq 2$ one of the generators of the mapping class group of $N_{g,n}$ is a crosscap transposition. We give explicit formulae for the action of crosscap transpositions and their inverses on the set of multicurves in $N_{g,n}$ in terms of generalized Dynnikov coordinates.