Loops in surfaces and star-fillings

TL;DR

This paper introduces a novel graph-based method for computing algebraic operations on homotopy classes of loops in surfaces, simplifying calculations of intersection numbers, Lie brackets, and cobrackets.

Contribution

It presents a new approach using surface fillings by graphs to compute key algebraic operations on loops, offering potential computational advantages.

Findings

Provides a graph-based framework for loop algebraic operations

Simplifies calculations of intersection numbers and Lie brackets

Introduces a new method for homotopy class analysis in surfaces

Abstract

We discuss a new approach to computing the standard algebraic operations on homotopy classes of loops in surfaces: the homological intersection number, Goldman's Lie bracket, and the author's Lie cobracket. Our approach uses fillings of the surfaces by certain graphs.