Palettes of Dehn colorings for spatial graphs and the classification of vertex conditions

纽结的 Dehn 染色最小数与 $\mathcal{R}$-调色板图

Minimum numbers of Dehn colors of knots and $\mathcal{R}$-palette graphs

本文研究 Dehn 染色下纽结的最小染色数，证明了任意奇素数 $p$ 的 Dehn $p$-可染纽结最小染色数下界，定义了用于寻找颜色集候选的 $\R$-调色板图，并确定 Dehn $5$-可染纽结的最小染色数为 $4$。

Dehn colorings and vertex-weight invariants for spatial graphs

In this paper, we study Dehn colorings for spatial graphs, and give a family of spatial graph invariants that are called vertex-weight invariants. We give some examples of spatial graphs that can be distinguished by a vertex-weight invariant, whereas distinguished by neither their constituent links nor the number of Dehn colorings.

Dehn coloring and the dimer model for knots

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…

Constructing Goeritz matrix from Dehn coloring matrix

Associated to a knot diagram, Goeritz introduced an integral matrix, which is now called a Goeritz matrix. It was shown by Traldi that the solution space of the equations with Goeritz matrix (precisely, unreduced Goeritz matrix called in his paper) as a coefficient matrix is isomorphic to the linear space consisting of the Dehn colorings for a knot. In this paper, we give a construction of a Goeritz matrix from a Dehn coloring matrix, from which Dehn colorings are induced. Moreover, if the knot diagram is prime, we give a purely algebraic construction of a Goeritz matrix from a Dehn coloring m…

A note on Dehn colorings and invariant factors

If $A$ is an abelian group and $ϕ$ is an integer, let $A(ϕ)$ be the subgroup of $A$ consisting of elements $a \in A$ such that $ϕ\cdot a=0$. We prove that if $D$ is a diagram of a classical link $L$ and $0=ϕ_0,ϕ_1,\dots,ϕ_{n-1}$ are the invariant factors of an adjusted Goeritz matrix of $D$, then the group $\mathcal{D}_{A}(D)$ of Dehn colorings of $D$ with values in $A$ is isomorphic to the direct product of $A$ and $A=A(ϕ_{0}),A(ϕ_1),\dots,A(ϕ_{n-1})$. It follows that the Dehn coloring groups of $L$ are isomorphic to those of a connected sum of torus links $T_{(2,ϕ_1)} \text{ }\# \text{ } \cd…