A note on Dehn colorings and invariant factors

Abstract

If $A$ is an abelian group and $\varphi$ is an integer, let $A(\varphi )$ be the subgroup of $A$ consisting of elements $a\in A$ such that $\varphi \cdot a=0$. We prove that if $D$ is a diagram of a classical link $L$ and $0={\varphi }_0,{\varphi }_1,\dots ,{\varphi }_{n-1}$ 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 $A=A({\varphi }_0),A({\varphi }_1),\dots ,A({\varphi }_{n-1})$. It follows that the Dehn coloring groups of $L$ are isomorphic to those of a connected sum of torus links $T_{(2,{\varphi }_1)}\#\cdots \#T_{(2,{\varphi }_{n-1})}$.

v1: 10 pages, 4 figures. v2: 9 pages, 3 figures. Further changes may be made before publication in the Journal of Knot Theory and its Ramifications