Some Baumslag-Solitar groups are two bridges virtual knots

TL;DR

This paper establishes necessary conditions for certain two-generator, one-relator groups to be virtual knot groups, and fully characterizes when Baumslag-Solitar groups are 2-bridge virtual knot groups, also proving they are not classical 2-bridge knot groups.

Contribution

It provides a complete characterization of Baumslag-Solitar groups as 2-bridge virtual knot groups and introduces necessary conditions for such groups to be virtual knot groups.

Findings

Baumslag-Solitar groups can be characterized as 2-bridge virtual knot groups.

Necessary conditions for two-generator, one-relator groups to be virtual knot groups.

Baumslag-Solitar groups are not 2-bridge classical knot groups.

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.