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 [R. E. Byrd, On the geometry of virtual knots, M.S. Thesis, Boise State University (2012)], 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, 𝒞, a module over the group ring of H1(S; ℤ). The authors previously defined the Laplacian modules ℒG,ℒG∗ and polynomials ΔG, ΔG∗ associated to a Tait graph G and its dual G∗, and showed that the pairs {ℒG,ℒG∗}, {ΔG, ΔG∗} are isotopy invariants of ℓ. The relationship between 𝒞 and the Laplacian modules is described and used to prove that ΔG and ΔG∗ are equal when S is a torus.

中文翻译：

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加厚表面中链接的分组演示

使用组合论证，我们证明了众所周知的结果，即 3 空间中的链接的 Wirtinger 和 Dehn 表示描述了同构群。链接的结果不正确ℓ在加厚的表面小号 × [0, 1]. 它们的精确关系，如 [RE Byrd，关于虚拟结的几何，博伊西州立大学 MS 论文 (2012)] 中给出的，在这里通过一个基本论点建立。当一个图表在小号为了ℓ可以是棋盘阴影，Dehn 表示自然会导致一个阿贝尔的“Dehn 着色组”，一个同位素不变量ℓ. 引入同源信息小号产生更强的不变量，𝒞, 群环上的一个模H1(小号; ℤ). 作者之前定义了拉普拉斯模块ℒG,ℒG*和多项式ΔG, ΔG*与 Tait 图相关联G及其对偶G*, 并表明对{ℒG,ℒG*},{ΔG, ΔG*}是同位素不变量ℓ. 之间的关系𝒞并且描述了拉普拉斯模块并用于证明ΔG和ΔG*相等时小号是一个圆环。