A note on Dehn colorings and invariant factors

TL;DR

This paper characterizes Dehn coloring groups of classical links using invariant factors of an adjusted Goeritz matrix, revealing their structure as products involving the abelian group and its subgroups.

Contribution

It establishes a precise isomorphism between Dehn coloring groups and a product involving invariant factors, linking link invariants to algebraic structures.

Findings

Dehn coloring groups are isomorphic to a product involving the abelian group and invariant factors.

Dehn coloring groups of links correspond to those of connected sums of torus links.

The structure of Dehn coloring groups can be explicitly described using the invariant factors of the Goeritz matrix.

Abstract

If $A$ is an abelian group and $\phi$ is an integer, let $A(\phi)$ be the subgroup of $A$ consisting of elements $a \in A$ such that $\phi \cdot a=0$. We prove that if $D$ is a diagram of a classical link $L$ and $0=\phi_0,\phi_1,\dots,\phi_{n-1}$ 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 $A=A(\phi_{0}),A(\phi_1),\dots,A(\phi_{n-1})$. It follows that the Dehn coloring groups of $L$ are isomorphic to those of a connected sum of torus links $T_{(2,\phi_1)} \text{ }\# \text{ } \cdots \text{ } \# \text{ } T_{(2,\phi_{n-1})}$.