Torus Knots
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The (5,4) torus knot was used by John Robinson as the initial idea for ORACLE , which is a stretched out version of the knot. In the picture of the (5,4) torus knot you will see many crossings. How many?
It was proved by Murasugi in 1991, that if p, q are coprime numbers with |p|>|q| then the (p,q) torus knot has a crossing number |p|(|q| -1), i.e it has a diagram with this number of crossings, and this number is minimal. Also Kronheimer and Mrowka proved in 1993 that the unknotting number of T(p,q) is (|p|-1)(|q|-1)/2 .
Torus knot T(p,q) with small p, q can be pretty. Here is the trefoil T(3,2) and also its mirror image T(3,-2).
The Trefoil .....and its Mirror Image
When the trefoil T(3,2) has the Torus added in, it can look very effective, by changing round the thickness of the torus and the knot. How difficult would this be to build with a ribbon and ring in the real world ?
Don't you think the ribbon would keep falling off and taking short cuts? It would take a great deal of sticky tape and patience to create the same picture!
There is another trefoil, the T(2,3).
It doesn't really look like the same knot though does it! However, if you made a replica out of string (with the ends tied of course) and fiddled enough, you should be able to prove it's the same! This picture shows the trefoil as a 2-bridge knot.
Even unknotted forms, such as T(1,3) are attractive
and indeed this gives the edge of ETERNITY .
We have also discovered a page describing how to construct Maple plots of Torus Knots at The Geometry Center (outside link), University of Minnesota.
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