Re: Illusion and Forthrightness in Wikipedia

[Vaughan Pratt <pratt@cs.stanford.edu>]

On 9/13/2010 12:02 PM, Timothy Porter wrote:
>   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!

Granted a (3,2)-torus knot is also a trefoil knot topologically, but how
does that help Fred?  With the standard parametrization the plan view
lacks the lumps bugging Fred but it has four crossings when Fred wants
to keep it at three.

What's the minimal modification to the standard parametrization
x + iy = (cos(qt) + 2)exp(ipt) (sticking with z = sin(qt))
that gives a smooth trefoil with 3 crossings, for (p,q) either of (2,3)
or (3,2)?

I have this feeling there ought to be something slicker than
(cos(qt) - 4)exp(ipt) - 3 exp(-it) with (p,q) = (2,3) (what I gave
before, reflected about x+y = 0) but I can't see it.

(I also don't know what -4 and -3 should generalize to for other than
(2,3), but Fred hasn't asked for that yet.)

Monoidal but not symmetric (trying to stay in scope here).

Vaughan


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