Multivariate Alexander colorings

We extend the notion of link colorings with values in an Alexander quandle to link colorings with values in a module $M$ over the Laurent polynomial ring $\Lambda_{\mu}=\mathbb{Z}[t_1^{\pm1},\dots,t_{\mu}^{\pm1}]$. If $D$ is a diagram of a link $L$ with $\mu$ components, then the colorings of $D$ with values in $M$ form a $\Lambda_{\mu}$-module $\mathrm{Color}_A(D,M)$. Extending a result of Inoue [Kodai Math.\ J.\ 33 (2010), 116-122], we show that $\mathrm{Color}_A(D,M)$ is isomorphic to the module of $\Lambda_{\mu}$-linear maps from the Alexander module of $L$ to $M$. In particular, suppose $M$ is a field and $\varphi:\Lambda_{\mu} \to M$ is a homomorphism of rings with unity. Then $\varphi$ defines a $\Lambda_{\mu}$-module structure on $M$, which we denote $M_\varphi$. We show that the dimension of $\mathrm{Color}_A(D,M_\varphi)$ as a vector space over $M$ is determined by the images under $\varphi$ of the elementary ideals of $L$. This result applies in the special case of Fox tricolorings, which correspond to $M=GF(3)$ and $\varphi(t_i) \equiv-1$. Examples show that even in this special case, the higher Alexander polynomials do not suffice to determine $|\mathrm{Color}_A(D,M_\varphi)|$; this observation corrects erroneous statements of Inoue [J. Knot Theory Ramifications 10 (2001), 813-821; op. cit.].