Tricolorability

Remarks

Each Reidemeister move turns a tricoloring into a tricoloring, so if one diagram of $K$ is tricolorable then every diagram of $K$ is. Example: the standard diagram of the trefoil knot has three arcs and three crossings, and each crossing meets all three arcs; three distinct colors satisfy both clauses. Non-example: the round diagram of the unknot has one arc, so clause 2 fails; hence the trefoil is not the unknot. The figure-eight knot is not tricolorable. With the colors read as elements of $\Z/3$, clause 1 says $2a \equiv b + c \pmod 3$ at a crossing with over-arc $a$ and under-arcs $b, c$; this is Fox $3$-coloring. A knot is tricolorable if and only if $3$ divides its determinant $\abs{\Delta_K(-1)}$, where $\Delta_K$ is the Alexander polynomial.