Geometric intersections of loops on surfaces

TL;DR

This paper introduces a new method combining Nielsen fixed point theory and Gröbner-Shirshov bases to efficiently compute geometric intersection and self-intersection numbers of loops on surfaces.

Contribution

It presents a novel approach that simplifies the calculation of intersection numbers using advanced algebraic and topological tools.

Findings

Provides a practical computational method for intersection numbers

Simplifies previous complex calculations

Applicable to various surface types

Abstract

Based on Nielsen fixed point theory and Gr\"{o}bner-Shirshov basis, we obtain a simple method to compute geometric intersection numbers and self-intersection geometric numbers of loops on surfaces.