Coloring link diagrams by Alexander quandles

TL;DR

This paper investigates how link diagrams can be colored using Alexander quandles, revealing conditions under which non-trivial colorings exist based on the Alexander polynomial of the link.

Contribution

It establishes new criteria linking the Alexander polynomial to the existence of non-trivial Alexander quandle colorings of links.

Findings

If the Alexander polynomial is zero, non-trivial colorings exist for any non-trivial Alexander quandle.

If the Alexander polynomial equals one, only trivial colorings are possible.

For non-zero, non-one Alexander polynomials, non-trivial colorings exist in a specific quotient Alexander quandle.

Abstract

In this paper, we study the colorability of link diagrams by the Alexander quandles. We show that if the reduced Alexander polynomial $\Delta_{L}(t)$ is vanishing, then $L$ admits a non-trivial coloring by any non-trivial Alexander quandle $Q$, and that if $\Delta_{L}(t)=1$, then $L$ admits only the trivial coloring by any Alexander quandle $Q$, also show that if $\Delta_{L}(t)\not=0, 1$, then $L$ admits a non-trivial coloring by the Alexander quandle $\Lambda/(\Delta_{L}(t))$.