From: lrudolph@panix.com (Lee Rudolph)
Subject: Re: Knot theory - colourability
Date: 6 Sep 1999 14:32:33 -0400
Newsgroups: sci.math

anna_gardiner@my-deja.com writes:

>Hiya Guys,
>Me again.  This time, I'm wondering if any of you
>know the rules for n-colourability on knots.
>
>I know the rule for n=3 is that you assign each
>colour a number and at each crossing the three
>colour numbers must be 0mod3.
>
>Anyone know either a general rule for any n.. or
>a rule for n=4.
>
>Failing that, anyone know a book that I can glean
>this info from?

The American Mathematical Society sells a videotaped lecture
by Sylvain Cappell of the Courant Institute, called "Coloring
Knots".  Here's a relevant extract from the review of it I
wrote for Mathematical Reviews (de-TeXed):

   Following Ralph Fox, [Cappell] develops the example
   of colorings of (diagrams of) knots, indicates why (say) 
   3-colorability and 5-colorability are knot invariants, 
   and thus is able to show by purely elementary combinatorial 
   means that the trefoil and the figure-8 are knotted, and the 
   Borromean rings linked; he also translates the combinatorics 
   into linear algebra (modulo p), reinterprets p-colorability 
   of K as existence of a representation of \pi_1(S^3\K) onto 
   the dihedral group D_p, and mentions the relevance of the 
   Alexander polynomial and the skein relations (which lead 
   to an offhand mention of the Jones polynomial).

A relevant (article in a) book, worth reading in any case for all
that it's going on 40 years old, is Fox's "A Quick Trip Through Knot
Theory" in _Topology of 3-Manifolds_ (ed. M. K. Fort).  No doubt
there are more recent sources (there may even be some articles
by Cappell in non-video format, but I can't make MathSci work
today, and I don't know any offhand).

Lee Rudolph
==============================================================================

From: Bob Riley <bob@math.binghamton.edu>
Subject: Re: Knot theory - colourability
Date: 6 Sep 1999 16:40:19 -0500
Newsgroups: sci.math

Lee Rudolph <lrudolph@panix.com> wrote:
> anna_gardiner@my-deja.com writes:

>>Hiya Guys,
>>Me again.  This time, I'm wondering if any of you
>>know the rules for n-colourability on knots.
>>
>>Anyone know either a general rule for any n.. or
>>a rule for n=4.
>>
>>Failing that, anyone know a book that I can glean
>>this info from?

   I don't have my possibly relevent books at hand, but
R H Crowell,
            Knots and Wheels
said to be in something called
                               N.C.T.M Yearbook (1961).
is what you want.  I can't find my copy but I remember its contents
wery well.  I've lectured on this several times.  Basically choose
an integer n > 2 and consider a wheel with n spokes with Euclidean
symmetry group the dihedral group D_n of order 2*n.  (I have in
mind *odd* n).  Consider a regular diagram of a knot: this has
components (arcs), each starting and ending as undercrossing at a
double point.  A symmetric representation of the knot on the wheel
is a function which assigns a spoke to each arc of the diagram, so
that: the flip (rotation of the wheel about an axis in the plane of
the wheel of order 2) about the spoke corresponding to an overcrossing
interchanges the spokes corresponding to the two undercrossing segments
at the crossing point.  If more than one spoke appears in the image
of the representation we really have a dihedral representation of the
knot group.  You convert this to n-colouring by painting each spoke
with a distinctive colour.  Crowell shows that the possibility of a
symmetric representation on an n spoked wheel using more than one
spoke is not changed by the three Reidemeister operations applied to
the diagram of the knot.  Why not take this as an exercise and prove
it yourself?

> The American Mathematical Society sells a videotaped lecture
> by Sylvain Cappell of the Courant Institute, called "Coloring
> Knots".  Here's a relevant extract from the review of it I
> wrote for Mathematical Reviews (de-TeXed):

>    the dihedral group D_p, and mentions the relevance of the 
>    Alexander polynomial and the skein relations (which lead 
>    to an offhand mention of the Jones polynomial).

   Somehow Crowell missed this last.

> A relevant (article in a) book, worth reading in any case for all
> that it's going on 40 years old, is Fox's "A Quick Trip Through Knot
> Theory" in _Topology of 3-Manifolds_ (ed. M. K. Fort).  No doubt
> Lee Rudolph

   This I do have at hand.  It identifies the N.C.T.M Yearbook as
National Council of Teachers of Mathematics Yearbook.  Thanks Lee for
solving the mystery.

                                 R^2