posted on 2023-06-09, 14:47authored byAndrew Bartholomew, Roger Fenn, Naoko Kamada, Seiichi Kamada
Doodles were introduced in but were restricted to embedded circles in the 2-sphere. Khovanov, extended the idea to immersed circles in the 2-sphere. In this paper we further extend the range of doodles to any closed oriented surfaces. Uniqueness of minimal representatives is proved, and various example 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.