Group Presentations For Links In Thickened Surfaces

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 l in a thickened surface S x [0, 1]. 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 S for l can be checkerboard shaded, the Dehn presentation leads naturally to an abelian "Dehn coloring group," an isotopy invariant of l. Introducing homological information from S produces a stronger invariant, C, a module over the group ring of H-1(S; Z). The authors previously defined the Laplacian modules L-G, L-G* and polynomials Delta(G), Delta(G*) associated to a Tait graph G and its dual G*, and showed that the pairs {L-G, L-G*}, {Delta(G), Delta(G*)} are isotopy invariants of l. The relationship between C and the Laplacian modules is described and used to prove that Delta(G) and Delta(G*) are equal when S is a torus.