An excursion on doodles on surfaces and virtual twins

Study of certain isotopy classes of a finite collection of immersed circles without triple or higher intersections on closed oriented surfaces can be thought of as a planar analogue of virtual knot theory where the genus zero case corresponds to classical knot theory. Alexander and Markov theorems for the classical setting is well-known, where the role of groups is played by twin groups, a class of right-angled Coxeter groups with only far commutativity relations. In the talk, Alexander and Markov theorems for higher genus case, where the role of groups is played by a new class of groups called virtual twin groups, will be discussed which is work in collaboration with Dr Mahender Singh. Furthermore, recent work on structural aspects of these groups will be addressed which is a joint work with Dr Mahender Singh and Dr Tushar Kanta Naik.