Dehn coloring and the dimer model for knots
Dehn coloring and the dimer model for knots
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Dehn 着色和结的二聚体模型
DOI：
10.1142/s0218216517410085
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
A. Yasuhara
中科院分区：
文献类型：
--
作者：
Y. Ishii;A. Yasuhara
狐狸染色为研究纽结群的二面体表示提供了一个组合框架。不太为人所知的Dehn着色概念也捕捉到了同样的数据。卡特-西尔弗-威廉姆斯最近的工作澄清了两者之间的关系，重点是一个人如何在福克斯和德恩颜色之间转换。在我们的工作中，我们将Dehn染色与纽结的二聚体模型联系起来，表明Dehn染色数据是由某个加权平衡覆盖Tait图编码的。利用Kasteleyn理论，给出了计算纽结的行列式和Smith标准形的图论方法。这些构造与考夫曼关于亚历山大多项式的状态和的工作密切相关。
Fox coloring provides a combinatorial framework for studying dihedral representations of the knot group. The less well-known concept of Dehn coloring captures the same data. Recent work of Carter-Silver-Williams clarifies the relationship between the two focusing on how one transitions between Fox and Dehn colorings. In our work, we relate Dehn coloring to the dimer model for knots showing that Dehn coloring data is encoded by a certain weighted balanced overlaid Tait graph. Using Kasteleyn theory, we provide graph theoretic methods for computing the determinant and Smith normal form of a knot. These constructions are closely related to Kauffman's work on a state sum for the Alexander polynomial.