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, $\mathcal{C}$, 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 ${\Delta }_G,{\Delta }_{G^{\ast }}$ associated to a Tait graph $G$ and its dual $G^{\ast }$, and showed that the pairs $\{{\mathcal{L}}_G,{\mathcal{L}}_{G^{\ast }}\}$, $\{{\Delta }_G,{\Delta }_{G^{\ast }}\}$ are isotopy invariants of $\ell$. The relationship between $\mathcal{C}$ and the Laplacian modules is described and used to prove that ${\Delta }_G$ and ${\Delta }_{G^{\ast }}$ are equal when $S$ is a torus.

16 pages, 12 figures