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  "body": "\n## Definition\n\nLet $D$ be a diagram of a [[knot|QYEAZTYGPT]] $K$.\nThe *arcs* of $D$ are the pieces into which $D$ falls when each under-strand is cut at each crossing.\nA *tricoloring* of $D$ is a map from the set of arcs of $D$ to a set of three colors such that:\n\n1. at each crossing, the over-arc and the two under-arcs have either one color or three distinct colors;\n2. at least two colors are used.\n\nThe diagram $D$ is *tricolorable* if it has a tricoloring.\nThe knot $K$ is *tricolorable* if some diagram of $K$ is tricolorable.\n\n## Remarks\n\nEach [[Reidemeister move|T3HAGWYKK7]] turns a tricoloring into a tricoloring, so if one diagram of $K$ is tricolorable then every diagram of $K$ is.\nExample: the standard diagram of the [[trefoil knot|VYTYSVHN4A]] has three arcs and three crossings, and each crossing meets all three arcs; three distinct colors satisfy both clauses.\nNon-example: the round diagram of the [[unknot|1TQ06R4Q66]] has one arc, so clause 2 fails; hence the trefoil is not the unknot.\nThe [[figure-eight knot|C01XBGAHC6]] is not tricolorable.\nWith 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.\nA knot is tricolorable if and only if $3$ divides its determinant $\\abs{\\Delta_K(-1)}$, where $\\Delta_K$ is the [[Alexander polynomial|JKA25KDPQP]].\n",
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