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 $\mathrm{\ell }$ 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 $\mathrm{\ell }$ can be checkerboard shaded, the Dehn presentation leads naturally to an abelian “Dehn coloring group,” an isotopy invariant of $\mathrm{\ell }$. Introducing homological information from $S$ produces a stronger invariant, $\mathcal{C}$, a module over the group ring of $H_1(S;\mathbb{Z})$. The authors previously defined the Laplacian modules ${\mathcal{L}}_G,{\mathcal{L}}_{G^{\ast }}$ and polynomials ${\mathrm{\Delta }}_G,{\mathrm{\Delta }}_{G^{\ast }}$ associated to a Tait graph $G$ and its dual $G^{\ast }$, and showed that the pairs $\{{\mathcal{L}}_G,{\mathcal{L}}_{G^{\ast }}\}$, ${\Delta_G, \Delta_{G^}}$areisotopyinvariantsof$\ell$\mathrm{.}Therelationshipbetween$\cal C$andtheLaplacianmodulesisdescribedandusedtoprovethat$\Delta_G$and$\Delta_{G^}$areequalwhen$S$ is a torus.