Doodles and commutator identities

Abstract

A doodle is a collection of immersed circles without triple intersections in the 2-sphere. It was shown by the second author and 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.