Fox n-coloring

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In the mathematical field of knot theory, Fox n-coloring is a method for obtaining invariants of links by coloring arcs in a link diagram. It is named after Ralph Fox, who invented it (and the special case of tricolorability) around 1960. It has been popularized by Jose Montesinos and Louis Kauffman.

The n-coloring of a link diagram D is an assignment of each arc of the diagram to one of n colors, considered as elements of $\mathbb Z_n$ , such that at every crossing the sum of the undercrossing colors is equal to twice the color of the overcrossing.

For the special case of tricoloring (n = 3), this is the same as requiring:

Arcs may only change colors when passing under another arc at a crossing
At any crossing only one or three unique colors can be present

The number of unique Fox n-colorings of a link diagram D, denoted by

col n (D),

is actually an invariant of the link represented by the diagram, as it is preserved under the Reidemeister moves. For tricolorings, the number of colorings is denoted by

tri(D) = col 3 (D)

All possible tricolorings of the trefoil knot.

For example, the standard minimal crossing diagram of the Trefoil knot has 9 distinct tricolorings as seen in the figure:

3 "trivial" colorings (every arc blue, red, or green)
3 colorings with the ordering Blue->Green->Red
3 colorings with the ordering Blue->Red->Green

With this fact, one can prove that the trefoil is not a trivial knot, by comparing its tricolorings with those of the trivial knot

$\mathrm{tri}(\mathrm{trefoil}) = 9 \neq 3 = \mathrm{tri}(\mathrm{trivial})$ .

A coloring where only one color is used is called a trivial coloring. Note every knot diagram admits a trivial coloring. A knot that admits a nontrivial tricoloring, i.e. one where all three colors are used, is often called tricolorable. See Tricolorability for more information.