  Given a knot K in an Euclidean space E and a finite dimensional space V of
smooth functions on K, we express the expected number of critical points of a
random function in V in terms of an integral-geometric invariant of K and V.
When V consists of the restrictions to K of homogeneous polynomials of degree d
on E, this invariant takes the form of the total curvature of a certain
immersion of K. In particular, when K is the unit circle in the plane centered
at the origin, then the expected number of critical points of the restriction
to K of a random homogeneous polynomial of degree d is $2\sqrt{3d-2}$, and the
expected number of critical points on K of a trigonometric polynomial of degree
d is approximately 1.549d.
