Narrow Results

Giving a presentation of the group of a 2-braid virtual knot or link, we consider the groups of three families of 2-braid virtual knots. Each of them has a certain feature; for example, we can show: for any positive integer N, there exists a virtual knot group with an element of order N. It is known that the collection of the virtual knot groups is the same as that of the ribbon T²-knot groups. Using our examples we discuss the relationship among the virtual knot groups and other knot groups such as ribbon S²-knot groups, S²-knot groups, T²-knot groups, and S³-knot groups.

A ribbon torus knot embedded in the 4-space is presented by a welded virtual knot through the tube operation due to Shin Satoh. We make an attempt of classification of ribbon torus knots presented by virtual knots with up to four crossings, where we use the list of virtual knots enumerated by Jeremy Green. We mainly investigate the groups of virtual knots.