论文标题
在增厚表面中链接的小组演示
Group Presentations for Links in Thickened Surfaces
论文作者
论文摘要
使用组合参数，我们证明了众所周知的结果，即三个空间中的链接的电线和DEHN演示描述了同构基团。对于links $ \ ell $，在增厚的表面$ s \ times [0,1] $中的链接不正确。正如2012年R.E.伯德（Byrd）在这里建立了一个基本论点。当可以为$ \ ell $的$ s $中的一个图表造成棋盘上的阴影时，Dehn演示文稿自然而然地导致了Abelian“ Dehn Coloring Group”，这是$ \ ell $的同位素不变的。从$ s $中引入同源信息会产生一个更强大的不变性，$ \ cal c $，一个$ h_1（s; {\ mathbb z}）$的组环上的模块。作者先前定义了Laplacian模块$ {\ cal l} _g，{\ cal l} _ {g^*} $和多项式$δ_g，δ__{g^*} $与Tait Gragr $ G $相关l} _ {g^*} \} $，$ \ {δ_g，δ__{g^*} \} $是$ \ ell $的同位素不变。描述了$ \ cal c $和拉普拉斯模块之间的关系，并用来证明$Δ_g$和$δ__{g^*} $是当$ s $是圆环时相同的。
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 $Δ_G, Δ_{G^*}$ associated to a Tait graph $G$ and its dual $G^*$, and showed that the pairs $\{{\cal L}_G, {\cal L}_{G^*}\}$, $\{Δ_G, Δ_{G^*}\}$ are isotopy invariants of $\ell$. The relationship between $\cal C$ and the Laplacian modules is described and used to prove that $Δ_G$ and $Δ_{G^*}$ are equal when $S$ is a torus.
