Multivariate Alexander colorings

TL;DR

This paper generalizes link colorings using Alexander quandles to modules over Laurent polynomial rings, establishing a module isomorphism with Alexander modules and analyzing coloring dimensions over fields, with implications for knot invariants.

Contribution

It extends Alexander coloring theory to modules over Laurent polynomial rings and relates coloring modules to Alexander modules, correcting previous misconceptions.

Findings

Coloring modules are isomorphic to homomorphism modules from Alexander modules.

The dimension of colorings over a field depends on images of elementary ideals.

Higher Alexander polynomials do not fully determine coloring counts.

Abstract

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.].