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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