nLab
colorable knot

Colourability

Idea

The colourability of a knot tells one information about its knot group yet has a simple, and visually attractive aspect that seems almost to avoid all mention of groups, presentations, etc., except at a fairly naive level.

The easiest form of colourability to examine is 33-colourability.

33-colourability

Definition

A knot diagram is 33-colourable if we can assign colours to its arcs such that

  1. each arc is assigned one colour;

  2. exactly three colours are used in the assignment;

  3. at each crossing, either all the arcs have the same colour, or arcs of all three colours meet in the crossing.

Examples and non-examples
  • The usual diagram for the trefoil knot is 33-colourable. (Just do it! Each arc is given a separate colour and it works.)

  • The figure-8 knot diagram

    is not 33-colourable. (Try it!)

Theorem

33-colourability is a knot invariant.

The proof is amusing to work out oneself. You have to show that if a knot diagram DD is 33-colourable and you perform a Reidemeister move on it then the result is also 33-colourable. The thing to note is that any arcs that leave the locality of the move must be coloured the same before and after the move is done.

Notes
  • We can now use phrases such as ‘the trefoil knot is 33-colourable’ as its validity does not depend on what diagram is used to represent it, (by the above and by Reidemeister's theorem.)

  • As the trefoil knot is 33-colourable and the unknot is not, non-trivial knots exist. Moreover, the trefoil is 33-colourable and the figure 88 is not, so these are different. We also get that the bridge number of the trefoil is 22, as this provides the missing piece of the argument found in that entry.

There are two comments to make here. First what does this all mean at a deeper topological level? The other is : why stop at 33? What about nn-colourability? We will handle the second one first.

nn-colourability

Let nn be an integer - in practice n=1n = 1 or 2 are not that interesting, so usually n3n \geq 3. Let n\mathbb{Z}_n be the additive group of integers modulo nn.

Definition

An nn-colouring of a knot diagram, DD, is an assignment to each arc of an element of n\mathbb{Z}_n in such a way that, at each crossing, the sum of the values assigned on the underpass arcs is twice that on the overpass, and such that in the assignment at least two elements of n\mathbb{Z}_n are used.

<inkscape:perspective id='perspective10' inkscape:persp3d-origin='372.04724 : 350.78739 : 1' inkscape:vp_x='0 : 526.18109 : 1' inkscape:vp_y='0 : 1000 : 0' inkscape:vp_z='744.09448 : 526.18109 : 1' sodipodi:type='inkscape:persp3d'></inkscape:perspective> <sodipodi:namedview bordercolor='#666666' borderopacity='1.0' gridtolerance='10000' guidetolerance='10' id='base' inkscape:current-layer='layer1' inkscape:cx='47.132334' inkscape:cy='42.148486' inkscape:document-units='px' inkscape:pageopacity='0.0' inkscape:pageshadow='2' inkscape:window-height='701' inkscape:window-width='640' inkscape:window-x='0' inkscape:window-y='22' inkscape:zoom='2.8' objecttolerance='10' pagecolor='#ffffff' showgrid='false'></sodipodi:namedview> <rdf:RDF> <cc:Work rdf:about=''> <dc:format>image/svg+xml</dc:format> <dc:type rdf:resource='http://purl.org/dc/dcmitype/StillImage'></dc:type> </cc:Work> </rdf:RDF> a c b

with a+c n2ba+c \equiv_n 2b.

=– What, of course, needs to be checked (left to ‘the reader’) is

  • this notion is a generalisation of 3-colourability (i.e. coincides in the case of n=3n = 3);

  • nn-colourability is preserved under Reidemeister moves so can be applied to a knot, not just to a knot diagram;

and then to find some examples of, say, 5-colourability. What knots are 5-colourable? Which (2,k)(2,k)-torus knots are 5-colorable, and so on. This is not important mathematically, but is quite fun and, in fact, does give insight and experience in working with the Reidemeister moves.

Coloring by a quandle

There is an even more general notion of coloring a knot KK by the elements of a quandle (Q,:Q×QQ)(Q,\rhd : Q \times Q \to Q). Formally, a coloring of KK by QQ corresponds to a quandle homomorphism from the fundamental quandle Q(K)Q(K) of KK to QQ. Concretely, this says that at each crossing with arcs labelled aa, bb, and cc (as in the above diagram), the identity c=abc = a \rhd b must be respected. In particular, nn-coloring corresponds to coloring by the set n\mathbb{Z}_n equipped with the quandle operation ab=2ba(modn)a \rhd b = 2b - a\,(mod\,n), known as the dihedral quandle.

References

category: knot theory

Revised on April 16, 2014 03:00:32 by Noam Zeilberger (195.83.213.132)