2020/05/04 by Silver, Daniel S., Williams, Susan G.
#05C10 #57M25 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2005.01576
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 × [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 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, \cal C, a module over the group ring of H1(S; \mathbb Z). The authors previously defined the Laplacian modules \cal LG, \cal LG^* and polynomials ΔG, ΔG^* associated to a Tait graph G and its dual G^*, and showed that the pairs \\cal LG, \cal LG^*\, \ΔG, ΔG^*\ are isotopy invariants of ℓ. The relationship between \cal C and the Laplacian modules is described and used to prove that ΔG and ΔG^* are equal when S is a torus.