View source for Fox-n-colouring

{{MSC|57K}}

A colouring of a non-oriented link diagram (cf. also [[Knot and link diagrams]]), leading to an Abelian group invariant of links in $\RR^3$ (cf. also [[Link|Link]]). It was introduced by R.H. Fox around 1956 to visualize dihedral representations of the knot group [[#References|[a1]]] (cf. also [[Knot and link groups]]). Using $3$-colourings is, probably, the simplest method of showing that the trefoil knot is non-trivial (see Fig. a1).

One says that a link (or tangle) diagram, $D$, is $n$-coloured if every arc is coloured by one of the numbers $0,\ldots,(n-1)$ in such a way that at each crossing the sum of the colours of the undercrossings is equal to twice the colour of the overcrossing modulo $n$. The set of $n$-colourings forms an [[Abelian group]], denoted by $\text{Col}_n(D)$. This group can be interpreted using the first homology group (modulo $n$) of the double branched cover of $S^3$ with the link as the branched point set. The group of $3$-colourings is determined by the [[Jones polynomial]] (at $t=e^{2\pi i/6}$), and the group of $5$-colourings by the [[Kauffman polynomial]] (at $a=1$, $z = 2\cos(2\pi/5)$), [[#References|[a2]]]. The $n$-moves preserve the group of $n$-colourings and $3$-moves lead to the [[Montesinos–Nakanishi conjecture]] (cf. [[Tangle move]]).

[[File:Fox coloring.svg|center|130px|Fox coloring with {0,1,2}]]

The linear space of $p$-colourings of the boundary points of an $n$-[[tangle]] has a symplectic form (cf. also [[Symplectic structure]]), so that tangles correspond to Lagrangian subspaces (i.e. maximal totally degenerate subspaces) of the symplectic form.

The Alexander module is a generalization of the group of $n$-colourings.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  R.H. Crowell,   R.H. Fox,   "An introduction to knot theory" , Ginn  (1963) {{ZBL|0126.39105}}.  4th corr. print. Graduate Texts in Mathematics '''57''', Springer (1977) {{ISBN|3-540-90272-4}} {{ZBL|0362.55001}}.  Repr. Dover (2008) {{ISBN|0-486-46894-1}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Przytycki,  ''3-coloring and other elementary invariants of knots'', in "Knot Theory" (ed. V.F.R. Jones) Banach Center Publications '''42'''  (1998)  pp. 275–295 [http://eudml.org/doc/208812 EUDML] {{ZBL|0904.57002}}</TD></TR>
</table>

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