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A Knot Game *January 14, 2009*

*Posted by keithkchan in fun stuffs.*

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Yesterday my roommate Mr Zurab gave me a quiz. Hold a string in your hands, and there is no knot on the wire as follows

That you can twist your body in whatever way you like but there is no knot in the string.

Now your task is to manipulate the string and/or your body so that there is a knot in the string. I give the answer in the comment. Don’t look at it immediately.

Why is this interesting? Recently I read Dmitry Podolsky’s post, in which he talked about a nice introduction to anyon by Frank Wilczek . In path integral, in between two end points, there are infinite many possible histories. The paths can be classified into topologically different paths. For example whether the paths cross each other or not. For boson, those paths just add, for fermions, they subtract. In 3D, the topology of the paths are simple since there is a transverse direction for two paths to bypass each other. In 2D, the topology, it is much more tricky and there are more interesting possibilities, which give rise to the possibility of anyons.

The connection with the knot is that the classification of different classes of paths is similar to analyze the structure of knots.

With your hands twisted so that a knot is formed by your hands. Then you translate the knot in your hands to the string.

The number of knots in the string is a topologically invariant. So in a loop there is no knot at the beginning, you can generate knots without twisting the string across each.

We also verify there is no cancellation of knots. That is, when you can make a knot on one side, another knot on the other side, move them towards each other, they will not cancel each other.