Some Baumslag-Solitar groups are two bridges virtual knots

Abstract

In this paper we give necessary conditions on group presentations, with two generators and one relator, in order to be the group of a virtual knot diagram. Although those conditions are not enough, we use them to determine, completely, whether or not a Baumslag-Solitar group is the group of a $2$-bridge virtual knot. Moreover, we present a combinatorial proof of the fact that these groups are not $2$-bridge classical knot groups.