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In the mathematical field of knot theory, Fox n-coloring is a method of specifying a representation of a knot group or a group of a link (not to be confused with a link group) onto the dihedral group of order n where n is an odd integer by coloring arcs in a link diagram (the representation itself is also often called a Fox n-coloring). Ralph Fox discovered this method (and the special case of tricolorability) 'in an effort to make the subject accessible to everyone' when he was explaining knot theory to undergraduate students at Haverford College in 1956. Fox n-coloring is an example of a conjugation quandle. Definition Let L be a link, and let {\displaystyle \pi } be the fundamental group of its complement. A representation {\displaystyle \rho } of {\displaystyle \pi } onto D 2 n {\displaystyle D_{2n}} the dihedral group of order 2n is called a Fox n-coloring (or simply an n-coloring) of L. A link L which admits such a representation is said to be n-colorable, and {\displaystyle \rho } is called an n-coloring of L. Such representations of groups of links had been considered in the context of covering spaces since Reidemeister in 1929. [Actually, Reidemeister fully explained all this in 1926, on page 18 of 'Knoten und Gruppen' in Hamburger Abhandlungen 5. ]The group of a link is generated by paths from a basepoint in S 3 {\displaystyle S^{3}} to the boundary of a tubular neighbourhood of the link, around a meridian of the tubular neighbourhood, and back to the basepoint. By surjectivity of the representation these generators must map to reflections of a regular n-gon. Such reflections correspond to elements t s i {\displaystyle ts^{i}} of the dihedral group, where t is a reflection and s is a generating ( 2 / n {\displaystyle 2\pi /n} ) rotation of the n-gon. The generators of the group of a link given above are in bijective correspondence with arcs of a link diagram, and if a generator maps to t s i D 2 p {\displaystyle ts^{i}\in D_{2p}} we color the corresponding arc i Z / p Z {\displaystyle i\in \mathbb {Z} /p\mathbb {Z} }. This is called a Fox n-coloring of the link diagram, and it satisfies the following properties:At least two colors are used (by surjectivity of {\displaystyle \rho } ).
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