  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.
