Doodles and commutator identities

TL;DR

This paper explores the relationship between doodles—collections of immersed circles on a sphere—and commutator identities in free groups, establishing a bijection between doodle cobordism classes and elementary commutator identities.

Contribution

It provides a detailed analysis of doodles with noose systems and elementary identities, revealing a bijection with weak equivalence classes of commutator identities.

Findings

Bijection between cobordism classes of colored doodles and elementary commutator identities.

Characterization of doodles with noose systems in terms of commutator identities.

Deeper understanding of the algebraic-topological correspondence in free groups.

Abstract

A doodle is a collection of immersed circles without triple intersections in the $2$-sphere. It was shown by the second author and P.~Tayler that doodles induce commutator identities (identities amongst commutators) in a free group. In this paper we observe this idea more closely by concentrating on doodles with proper noose systems and elementary commutator identities. In particular we show that there is a bijection between cobordism classes of colored doodles and weak equivalence classes of elementary commutator identities.