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
Read more about this topic: Fox N-coloring
Famous quotes containing the words number of and/or number:
“A good marriage ... is a sweet association in life: full of constancy, trust, and an infinite number of useful and solid services and mutual obligations.”
—Michel de Montaigne (1533–1592)
“Take away from the courts, if it could be taken away, the power to issue injunctions in labor disputes, and it would create a privileged class among the laborers and save the lawless among their number from a most needful remedy available to all men for the protection of their business interests against unlawful invasion.... The secondary boycott is an instrument of tyranny, and ought not to be made legitimate.”
—William Howard Taft (1857–1930)