Coloring link diagrams by Alexander quandles

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 ${\Delta }_L(t)\ne 0,1$, then $L$ admits a non-trivial coloring by the Alexander quandle $\Lambda /({\Delta }_L(t))$.

13 pages, 1 figure