Virtual knot groups (Q2725562)

Virtual knots can be defined combinatorially by diagrams with two kinds of crossings, up to an extended set of Reidemeister moves. In this paper the authors define the group of a virtual knot, using an analogue of Wirtinger's presentation and remark that such presentation need not have deficiency 1, unlike the classical case. They give necessary and sufficient conditions for a group \(G\) to be virtual knot group, both in combinatorial terms, and as the fundamental group of the complement of a ribbon torus in \(\mathbb{R}^4\). They observe that virtual knot groups need not be residually finite.NEWLINENEWLINENEWLINEThey go on to extend colourings of a diagram by a continuous palette of colours, which they had studied in previous papers, to the virtual case. They are then able to identify simple moves on a virtual diagram which leave some of these colourings unaltered, while potentially altering the virtual knot.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00034].