Virtual twin groups and doodles on surfaces

Abstract: The 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 genus zero case are known, where the role of groups is played by twin groups, a class of right-angled Coxeter groups with only far commutativity relations. In this talk, I will discuss Alexander and Markov theorems for the higher genus case, where the role of groups is played by a new class of groups called virtual twin groups. This is joint work with Dr Mahender Singh. I will also address recent work on the structural aspects of these groups, which is joint work with Dr Mahender Singh and Dr Tushar Kanta Naik.

group theory

Audience: researchers in the topic

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