Combinatorial Lie bialgebras of curves on surfaces

TL;DR

This paper introduces a combinatorial algorithm to compute the Goldman-Turaev Lie bialgebra structure on curves on surfaces, generalizes Goldman's results, and explores characterizations of simple curves, providing new insights and counterexamples.

Contribution

It presents a novel combinatorial method for computing the Lie bialgebra of curves and extends Goldman's theorem, also addressing Turaev's conjecture with new examples and questions.

Findings

Algorithm for computing the Lie bialgebra structure

Counterexamples to Turaev's characterization of simple curves

Proposed alternative characterization of simple curves

Abstract

Goldman and Turaev found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of curves on a surface. When the surface has non-empty boundary, this vector space has a basis of cyclic reduced words in the generators of the fundamental group and their inverses. We give a combinatorial algorithm to compute this Lie bialgebra on this vector space of cyclic words. Using this presentation, we prove a generalization of Goldman's result relating the bracket to disjointness of curve representatives when one of the classes is simple. We exhibit some examples we found by programming the algorithm which answer negatively Turaev's question about the characterization of simple curves in terms of the cobracket. Further computations suggest an alternative characterization of simple curves in terms of the bracket of a curve and its inverse. Turaev's question is still open in genus zero.