Minimally intersecting filling pairs on the punctured surface of genus two

TL;DR

This paper constructs a minimally intersecting pair of simple closed curves filling a genus 2 surface with an odd number of punctures greater than 3, completing the classification for all such surfaces.

Contribution

It provides the explicit construction for minimally intersecting filling pairs on genus 2 surfaces with certain punctures, finishing prior classification efforts.

Findings

Constructed minimally intersecting filling pairs for genus 2 surfaces with odd punctures > 3

Completed the classification of such pairs for all surfaces

Advances understanding of curve configurations on punctured surfaces

Abstract

In this short note, we construct a minimally intersecting pair of simple closed curves that fill a genus 2 surface with an odd, greater than 3, number of punctures. This finishes the determination of minimally intersecting filling pairs for all surfaces completing the work of Aougab-Huang and Aougab-Taylor.