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 Subject: Re: Knot theory - colourability Date: 6 Sep 1999 16:40:19 -0500 Newsgroups: sci.math Lee Rudolph 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