
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 over the Laurent polynomial ring . If is a diagram of a link with components, then the colorings of with values in form a -module . Extending a result of Inoue [Kodai Math.\ J.\ 33 (2010), 116-122], we show that is isomorphic to the module of -linear maps from the Alexander module of to . In particular, suppose is a field and is a homomorphism of rings with unity. Then defines a -module structure on , which we denote . We show that the dimension of as a vector space over is determined by the images…
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