签名模与二面体共轭拟阵计数不变量
Signature Modules and the Dihedral Conjugation-Quandle Counting Invariant
- Emmaus High School（Emmaus高中）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出利用二面体群反射与旋转签名构造签名模，通过Smith标准形计算共轭拟阵着色，证明其在Reidemeister移动下不变，并推广到广义二面体群。
AI中文摘要：
我们利用二面体群的反射与旋转签名，发展了一种Smith标准形方法，用于计算所有二面体群上的共轭拟阵着色。设$L = L_1 \cup \cdots \cup L_\ell$为具有链环分支$L_1, \ldots, L_\ell$的链环，并设$\operatorname{Conj}(D_n)$为二面体群$D_n$的共轭拟阵。我们利用半直积结构$D_n \cong \mathbb{Z}_n \rtimes \mathbb{Z}_2$将着色分离为分量签名。该分量签名告诉我们每个链环分支是由旋转还是反射着色，且指数在模$n$下分别赋值。一旦确定了分量签名，每个交叉关系就变成一个确定整数矩阵$M_\tau$的方程。阿贝尔群$A_\tau(L)$（我们称之为签名模）由整数矩阵$M_\tau$表示。我们证明了自由秩和非单位Smith标准形元素（等价地，$A_\tau(L)$的同构类）在Reidemeister移动下保持不变。完整的签名模族也决定了所有$D_n$的完整计数不变量族。全反射系统与Fox-$n$着色相同，而混合旋转-反射签名包含的信息给出了更强的不变量。我们还证明了每个签名模是多元Alexander模的$t_k = (-1)^{\tau_k}$特化，并将模的着色解释推广到每个阿贝尔群$B$的广义二面体群$\operatorname{Dih}(B)$。
英文摘要：
We use reflection and rotation signatures of dihedral groups to develop a Smith normal form method for computing the conjugation quandle colorings across all dihedral groups. Let $L = L_1 \cup \cdots \cup L_\ell$ be a link with link components $L_1, \ldots, L_\ell$, and let $\operatorname{Conj}(D_n)$ be the conjugation quandle of the dihedral group $D_n$. We use the semidirect-product structure $D_n \cong \mathbb{Z}_n \rtimes \mathbb{Z}_2$ to separate the coloring into a component signature. This component signature tells us whether each link component is colored by rotations or reflections, and the exponents are assigned separately under modulus $n$. Once the component signature is determined, every crossing relation becomes an equation that determines an integer matrix $M_τ$. The abelian group $A_τ(L)$, which we call the signature module, is represented by an integer matrix $M_τ$. We prove that the free rank and nonunit Smith normal form entries, or equivalently the isomorphism class of $A_τ(L)$, are preserved under the Reidemeister moves. The complete family of signature modules also determines the entire family of counting invariants for all $D_n$. The all-reflection system is the same as the Fox-$n$ colorings, while the mixed rotation-reflection signatures contain information that gives a stronger invariant. We also show that each signature module is the $t_k = (-1)^{τ_k}$ specialization of the multivariable Alexander module, and extend the coloring interpretation of the module to generalized dihedral groups $\operatorname{Dih}(B)$ for every abelian group $B$.
