Coloring link diagrams by Alexander quandles (Q2909495)

The main topic discussed in this article is colorability of link diagrams by Alexander quandles. It is well known that for a finite quandle \(X\), there is a one-to-one correspondence between quandle homomorphisms \(Q(K) \to X\) and colorings \(C : R \to X\). The focal point of this paper is the relationship between the reduced Alexander polynomial \(\Delta_L (t)\) and colorings by Alexander quandles.NEWLINENEWLINE The author presents a main theorem, Theorem 1.2 which states that for a link \(L\), \(\Delta_L (t)\) the reduced Alexander polynomial, we have that, {\parindent=6mm \begin{itemize}\item[1.] \(L\) admits a non-trivial coloring by any non-trivial Alexander quandle \(Q\) if \(\Delta_L (t) =0\) \item[2.] \(L\) admits only the trivial coloring by any Alexander quandle \(Q\) if \(\Delta_L (t) =1\) \item[3.] \(L\) admits a non-trivial coloring by the Alexander quandle \(\Lambda /(\Delta_L (t))\) if \(\Lambda \Delta_L (t) \neq 0,1\). NEWLINENEWLINE\end{itemize}} In the later sections of this paper, the author provides the proof of each result in three separate theorems. Examples include the knot \(8_{15}\) and the trefoil knot \(3_1\) followed by a table of links with non-trivial colorings by the Alexander quandle \(\Lambda /(\Delta_L (t))\).