The parametrisation given corresponds to the symmetric placing of the trefoil on a torus, the fact that you get the `lumps' is an artifact of that. The knot parametrisation can neatly `turned around' changing the role of p and q. That is also a trefoil! Have fun, Take care. Don't tie yourself in knots :-) Tim On 13/09/2010 01:11, Fred E.J. Linton wrote:
These somewhat off-topic remarks are at least almost as connected with category theory as the study of knots is, as they have their origin in my reliance on, and subsequent disillusionment with, the Wikipedia site for basic information about a certain knot.
The knot? -- the familiar trefoil knot, aka (2,3)-torus knot. The offending Wikipedia page? -- http://en.wikipedia.org/wiki/Torus_knot . The problem?
A reader might be forgiven for expecting that when a page of mathematical text offers a parametrization, in the form
x = (2 + cos((q phi)/p))(cos(phi)) , y = (2 + cos((q phi)/p))(sin(phi)) , z = sin((q phi)/p) ,
of each (p,q)-torus knot, and then offers an illustration dubbed (2,3)-torus knot, that said illustration might have been produced by means of the p=2, q=3 instance of the parametrization given.
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