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Group presentations for links in thickened surfaces
Journal of Knot Theory and Its Ramifications ( IF 0.3 ) Pub Date : 2021-04-23 , DOI: 10.1142/s021821652150022x Daniel S. Silver 1 , Susan G. Williams 1
Journal of Knot Theory and Its Ramifications ( IF 0.3 ) Pub Date : 2021-04-23 , DOI: 10.1142/s021821652150022x Daniel S. Silver 1 , Susan G. Williams 1
Affiliation
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 H 1 ( S ; ℤ ) . The authors previously defined the Laplacian modules ℒ G , ℒ G ∗ and polynomials Δ G , Δ G ∗ associated to a Tait graph G and its dual G ∗ , and showed that the pairs { ℒ G , ℒ G ∗ } , { Δ G , Δ G ∗ } are isotopy invariants of ℓ . The relationship between 𝒞 and the Laplacian modules is described and used to prove that Δ G and Δ G ∗ are equal when S is a torus.
中文翻译:
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H 1 ( 小号 ; ℤ ) . 作者之前定义了拉普拉斯模块ℒ G , ℒ G * 和多项式Δ G , Δ G * 与 Tait 图相关联G 及其对偶G * , 并表明对{ ℒ G , ℒ G * } ,{ Δ G , Δ G * } 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Δ G 和Δ G * 相等时小号 是一个圆环。
更新日期:2021-04-23
中文翻译:
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