Group presentations for links in thickened surfaces

Abstract

Using a combinatorial argument, we prove the well-known result that the Wirtinger and Dehn presentations of a link in 3-space describe isomorphic groups. The result is not true for links [Formula: see text] in a thickened surface [Formula: see text]. Their precise relationship, as given in [R. E. Byrd, On the geometry of virtual knots, M.S. Thesis, Boise State University (2012)], is established here by an elementary argument. When a diagram in [Formula: see text] for [Formula: see text] can be checkerboard shaded, the Dehn presentation leads naturally to an abelian “Dehn coloring group,” an isotopy invariant of [Formula: see text]. Introducing homological information from [Formula: see text] produces a stronger invariant, [Formula: see text], a module over the group ring of [Formula: see text]. The authors previously defined the Laplacian modules [Formula: see text] and polynomials [Formula: see text] associated to a Tait graph [Formula: see text] and its dual [Formula: see text], and showed that the pairs [Formula: see text], [Formula: see text] are isotopy invariants of [Formula: see text]. The relationship between [Formula: see text] and the Laplacian modules is described and used to prove that [Formula: see text] and [Formula: see text] are equal when [Formula: see text] is a torus.