User:ChuckHG/Workbench

There are several notions of coloring of a knot or link.


 * The original and simplest is that of tricoloring a knot, that is to say one colors the strands of a regular knot diagram with three colors, say R,G,B, such that at each crossing the colors all agree or are all distinct. A tricoloring is trivial if it is constant (consisting of just one color).  Any diagram representing a trivial knot has only trivial tricolorings.  Consequently, if a knot diagram has a nontrivial (nonconstant) tricoloring then it must represent a nontrivial knot.  The trefoil knot has nontrivial tricolorings and hence is a nontrivial knot.


 * The second notion, due to Ralph Hartzler Fox is that of an n-coloring of a knot (or link).

(more to come)

Colorings with values in a group
Let G be a group. A G-coloring of a knot or link diagram L is the assignment of an element of G to the strands of L such that, at each crossing, if c is the element of G assigned to the overcrossing strand and if a and b are the elements of G assigned to the two undercrossing strands, then a c-1 b c-1 = 1; equivalently, a = c b-1 c or b = c a-1 c. If the diagram L'  is obtained from a Reidemeister move, then it is easy to give L'  a G-coloring which agrees with that of L outside a neighborhood of the move. From this it follows that equivalent links have compatible G-colorings and that a trivial knot has only constant G-colorings. If the group G is cyclic of order n, this concept reduces to that of Fox n-coloring. That this is a sharper concept than Fox n-coloring is given by the fact that the torus knot T(3,5) has only constant n-colorings, but for the group G = A5, T(3,5) has non-constant G-colorings.

For any knot or link diagram L there is a universal G-coloring group G(L) defined as the group generated by the strands of L modulo the relations given by x z-1 y z-1 = 1, where for some crossing of L,  z is the overcrossing strand, and x and y are the undercrossing strands. Clearly, this coloring group G(L) is an invariant of the ambient isotopy class of L. For a knot diagram K, G(K) is the free product of an infinite cyclic group and a reduced coloring group G0(K). In the case of the trefoil, this reduced coloring group is cyclic of order 3. For the torus knot T(3,5), G0(T(3,5)) is the binary icosahedral group (of order 120). It is not difficult to see that for any knot diagram K, G0(K) is isomorphic to the fundamental group of the 2-fold cyclic branched covering space of the 3-sphere, branched along K. If the resulting branched covering space were simply connected, then by the Poincar&eacute; conjecture (which is now a theorem), it would be the 3-sphere, and the non-trivial deck transformation from the covering would produce an involution of the 3-sphere fixing the branched curve. By the Smith conjecture (also now a theorem), this would have to be an unknotted curve, and in the orbit space of the involution, the image of the branched set (which is the original knot K) would have to be unknotted. Consequently, K is unknotted if and only if its reduced coloring group G0(K) is the trivial group.

If a link L is the boundary of a (not necessarily orientable) surface having m components (none of the components being closed), then G(L) has a free group of rank m as a quotient group.

The coloring group of a knot or link
The coloring group of a (tame) knot (or link) in 3-space is defined from a knot (or link) diagram K as follows: G(K) is the group generated by the strands of the K with relations of the form x z-1 y z-1 = 1 for each crossing whose overstrand at that crossing is z and whose understrands at that crossing are x and y. Any sequence of Reidemeister moves connecting the two diagrams K and L determines a canonical isomorphism of groups G(K) $$ \cong $$ G(L'').