Revision c37d8044-76b7-44c9-9a26-4f840c3af3d8 - MathOverflow

[1]:http://groupoids.org.uk/outofline/out-home.html
[2]:http://www.evl.uic.edu/hypercomplex/html/dirac.html
[3]:http://groupoids.org.uk/images/pentoil.jpg
[4]:http://groupoids.org.uk/images/pentw2.jpg
[5]:http://groupoids.org.uk/images/reln-crsm.jpg
[6]:http://groupoids.org.uk/popmath/cpm/exhib/knotexhib.html


The coffee mug trick is also called the Philipine Wine Trick and should be related to the Dirac String Trick, which you can find by a web search, for example [here][2] and also in my presentation [Out of Line][1], where rotations in 3-space are related to the Projective Plane. 

A knot trick, I am not sure you would call it magic, has been shown to children and academics in many places. It rquires a pentoil knot of width 20" made of copper tubing, about 7mm diameter (made by a university workshop) shown in the following diagram:

![pentoi;][3]

It also needs some nice flexible boating rope. The rope is wrapped round the $x,y$ pieces according to the rule 
$$R=xyxyxy^{-1}x^{-1}y^{-1}x^{-1}y^{-1} $$
and the ends tied together, as in the following picture:

![pentwrap2][4]

A member of the sudience is then invited to come up and manipulate the loop of rope off the knot, starting by turning it upside down. This justifies the rule $R=1$. Of course the rule is the relation for the fundamental group of complement of the pentoil, which can, for the right audience, be deduced from the relations at each crossing given by the diagram 

![relcross][5]

and can be easily demonstrated with the knot and rope. 

It is also of interest to have a copper trefoil around to compare the relations. One warning: the use of rope does not really model the fundamental group, so be careful with a demo for the figure eight knot! 

I did the demo for one teenager and he said:"Where did you get that formula?" This demo knot has been well travelled, for many different types of audience; on one occasion the airline lost my luggage with the rope and I had to ask the taxi from the airport to to stop at a hardware store for me to buy d=some clothesline. I devised this trick for an undergraduate course in knot theory in the late 1970s. 

You can also see a [Knot Exhibition][6] whose aim is to use the notion of knot to explain some basic methods of mathematics to the general public.