Pretzel链环的Quandle着色拟阵
Quandle coloring quivers of pretzel links
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中文总结 AI 辅助
本文系统研究pretzel链环的二面体quandle着色及着色拟阵，确定3-pretzel情形，并在n为素数时给出4-pretzel及一般m-pretzel链环的着色数和拟阵结构，附完整证明。
中文摘要 AI 辅助
本文利用二面体quandle $\mathbb{Z}_{n}$ 对pretzel链环的quandle着色及quandle着色拟阵进行了系统研究。首先，我们系统地考察了3-pretzel链环的所有可能着色，确定了每种情况下的不同着色数量及其quandle着色拟阵的结构。为了获得更一般的结论，我们根据同余方程组系数矩阵的性质对 $n$ 施加限制。因此，在 $n$ 为素数的情况下，我们研究了4-pretzel链环的quandle着色数及quandle着色拟阵。最后，结合4-pretzel链环的结果，我们在 $n$ 为素数的情况下严格推导了一般 $m$-pretzel链环的着色数和quandle着色拟阵，并给出了完整证明。
英文摘要
In this paper, we conduct a systematic study of quandle colorings and quandle coloring quivers for pretzel links using the dihedral quandle $\mathbb{Z}_{n}$. First, we systematically investigate all possible colorings of 3-pretzel links, determining the number of distinct colorings in each case as well as the structure of their quandle coloring quivers. In order to obtain more general conclusions, we impose restrictions on $n$ based on the properties of the coefficient matrix of the system of congruence equations. So we examine the number of quandle colorings and the quandle coloring quivers for 4-pretzel links in the case where $n$ is prime. Finally, Combining the results of 4-pretzel links we rigorously derive both the coloring numbers and quandle coloring quivers for general $m$-pretzel links in the case where $n$ is prime, with full proofs provided.
