Core groups (Q6590032)

In this paper, the authors take a group knot invariant defined independently by \textit{A. J. Kelly} [Groups from link diagrams, Ph.D. Thesis, Warwick University, Coventry (1991)] and \textit{M. Wada} [Topology 31, No. 2, 399--406 (1992; Zbl 0758.57008)] and extend it to virtual links. The group itself is defined with generators given by the arcs of a link, and relators defined in the following manner: if arcs \(a, b, c\) meet, with \(a\) the overcrossing arc, then we add the relator \(ab^{-1}ac^{-1}\). This is termed the \textit{arc core group}. Another group invariant may be defined using \textit{regions} in the plane, which, for a classical diagram, are the components of the compliement of the link diagram. The authors give an alternate definition for regions which applies to the virtual case. The resulting group is termed the \textit{regional core group}.\N\NIn Theorem 1.4, the authors show that for classical links, the regional core group is a free product of the arc core group with the integers.\N\NThe authors proceed to observe some consequences of this for the Wirtinger group and Dehn group of a (virtual) link. Of particular interest, many of the results apply to any virtual link that can be given a checkerboard coloring, i.e., to \textit{almost-classical} virtual links. This includes some of the results used to prove Theorem 1.4, such as Proposition 2.2, which shows the existence of a homomorphism from the arc core group to the regional core group for any almost classical link diagram.