Tag Archives: knot

Another project idea

If anyone is still in the market for a project idea, here is another one, related to our recent discussions about the rotation number.  Suppose that we integrate the absolute value of the curvature of a closed plane curve (with respect to arc length),

\[ \int |\kappa_\gamma| ds \]

Then the minimum value we can obtain is \(2\pi\), which is what we get for a convex planar Jordan curve.  What’s more, in this form the theorem is still good for curves in 3-dimensional space (remember that I explained in class that in 3-space one can’t give the curvature a definite sign, but of course that is not an issue if we only want to integrate its absolute value.

In 1950, John Milnor (then aged 19) proved that for a knotted loop in 3-dimensional space the lower bound for the total curvature is \(4\pi\) rather than \(2\pi\).  In other words, if a space curve has sufficiently small total curvature, it must have the topology of the unknot.  Milnor’s proof was published in the Annals of Mathematics which is probably the most distinguished mathematics journal in the world.  (The same result was proved independently and slightly earlier by Istvan Fary; it is known as the Fary-Milnor theorem.)

Explaining the proof would make a good project.  I’ve made a copy of the original article available here.