Doodles on surfaces

Abstract

Doodles were introduced in [R. Fenn and P. Taylor, Introducing doodles, in Topology of Low-Dimensional Manifolds, Lecture Notes in Mathematics, Vol. 722 (Springer, Berlin, 1979), pp. 37–43] but were restricted to embedded circles in the [Formula: see text]-sphere. Khovanov [M. Khovanov, Doodle groups, Trans. Amer. Math. Soc. 349 (1997) 2297–2315] extended the idea to immersed circles in the [Formula: see text]-sphere. In this paper, we further extend the range of doodles to any closed oriented surface. Uniqueness of minimal representatives is proved, and various examples of doodles are given with their minimal representatives. We also introduce the notion of virtual doodles, and show that there is a natural one-to-one correspondence between doodles on surfaces and virtual doodles on the plane.