Planar knots and related groups

Mahender Singh (IISER)

Study of stable 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 sphere case corresponds to the classical knot theory. It is intriguing to know which class of groups serves the purpose that Artin braid groups serve in classical knot theory. Khovanov proved that twin groups, a class of right angled Coxeter groups with only far commutativity relations, serves the purpose for the sphere case. In a recent work we showed that an appropriate class of groups called virtual twin groups fits into a virtual analogue of the planar knot theory. The talk will give an overview of some recent topological and group theoretic developments along these lines.