Multiple curves on punctured orientable surfaces

TL;DR

This paper introduces a method to describe multiple curves on punctured orientable surfaces using their geometric intersection numbers with embedded curves, aiding in understanding surface topology.

Contribution

It provides a new characterization of multiple curves on punctured surfaces through intersection numbers, enhancing topological analysis tools.

Findings

Characterization of multiple curves via intersection numbers

Application to orientable surfaces of genus g with punctures

Framework for analyzing surface topology

Abstract

We describe each multiple curve on the orientable surface of genus-$g$ with $n$ punctures and one boundary component by using this multiple curve's geometric intersection number with the embedded curves in this surface.