A note on Dehn colorings and invariant factors

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 $T_{(2,\phi_1)} \text{ }\# \text{ } \cdots \text{ } \# \text{ } T_{(2,\phi_{n-1})}$.