# 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|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.

## MATHEMATICAL THEMES

• Borromean Rings - what they are and why they don't exist!
• The Möbius Band - what one looks like, experiments to try, and a beautiful rotating golden one enabling you to really see what one looks like in 3D (this is optional as 90Kb).
• Bernard Morin and the Brehm Model - how Bernard Morin showed John Robinson the Brehm Model of the Möbius Band and how to make one!
• The Projective Plane - how to create and understand the projective plane when it is not possible to physically construct (this relates to the Brehm Model in the Projective Plane). Also the Dirac String Trick.
• Fibre Bundles - what they are, how to make them, and examples of them in John Robinson's Work.
• Knots and Links - Introduction to the subject of Knot Theory, includes history of the subject, and a rotating mathematically constructed Immortality (the trefoil being one of the most basic knots).
• Fractals 3 pages, introducing Fractals, considering iteration, The Sierpinski Gasket and the applications of the subject.

• Exhibition "Mathematics and Knots"
• Ronnie Brown's Homepage
• John Robinson's Symbolic Sculpture
• Brochure (140Kb), giving a guided Tour around the Symbolic Sculptures at the University of Wales, Bangor.
• About the Centre for the Popularisation of Mathematics