Partitions of cyclic words and Goldman-Turaev Lie bialgebra

TL;DR

This paper introduces a new combinatorial approach to defining the Goldman bracket and Turaev cobracket on the free module generated by homotopy classes of closed curves on a surface, using partitions of cyclic words.

Contribution

It provides a novel combinatorial framework based on cyclic word partitions to define the Goldman-Turaev Lie bialgebra operations, improving understanding of their structure.

Findings

New combinatorial definitions of Lie bialgebra operations

Enhanced understanding of cyclic word partitions

Potential applications to surface topology and algebraic structures

Abstract

The free $\mathbb{Z}$-module generated from the set of non-trivial homotopy classes of closed curves on an oriented surface has the structure of Lie bialgebra by two operations, the Goldman bracket and Turaev cobracket. M. Chas gave a combinatorial redefinition of these two operations through a natural identification of the homotopy classes of closed curves on the surface with the cyclic words in the generators and their inverses of the fundamental group of the surface. We present a new approach to give a combinatorial definition of the bracket and cobracket, focusing on the information given by the partitions of cyclic words.