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Study of certain isotopy classes of a finite collection of immersed circleswithout triple or higher intersections on closed oriented surfaces can bethought of as a planar analogue of virtual knot theory where the genus zerocase corresponds to classical knot theory. Alexander and Markov theorems forthe genus zero case are known where the role of groups is played by twingroups, a class of right angled Coxeter groups with only far commutativityrelations. The purpose of this paper is to prove Alexander and Markov theoremsfor higher genus case where the role of groups is played by a new class ofgroups called virtual twin groups which extends twin groups in a natural way.

Alexander and Markov theorems for virtual doodles