Number of Colorings
The number of distinct Fox n-colorings of a link L, denoted
is an invariant of the link, which is easy to calculate by hand on any link diagram by coloring arcs according to the coloring rules. When counting colorings, by convention we also consider the case where all arcs are given the same color, and call such a coloring trivial.
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
The set of Fox 'n'-colorings of a link forms an abelian group, where the sum of two n-colorings is the n-coloring obtained by strandwise addition. This group splits as a direct sum
- ,
where the first summand corresponds to the n trivial (constant) colors, and nonzero elements of summand correspond to nontrivial n-colorings (modulo translations obtained by adding a constant to each strand).
If is the connected sum operator and and are links, then
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