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 $\ell$ in a thickened surface $S \times [0,1]$. Their precise relationship, as given in the 2012 thesis of R.E. Byrd, is established here by an elementary argument. When a diagram in $S$ for $\ell$ can be checkerboard shaded, the Dehn presentation leads naturally to an abelian "Dehn coloring group," an isotopy invariant of $\ell$. Introducing homological information from $S$ produces a stronger invariant, $\cal C$, a module over the group ring of $H_1(S; {\mathbb Z})$. The authors previously defined the Laplacian modules ${\cal L}G,{ \cal L}{G^*}$ and polynomials $\Delta_G, \Delta_{G^*}$ associated to a Tait graph $G$ and its dual $G^*$, and showed that the pairs ${{\cal L}G, {\cal L}{G^*}}$, ${\Delta_G, \Delta_{G^*}}$ are isotopy invariants of $\ell$. The relationship between $\cal C$ and the Laplacian modules is described and used to prove that $\Delta_G$ and $\Delta_{G^*}$ are equal when $S$ is a torus.