Statistics of linear families of smooth functions on knots

Details Statistics of linear families of smooth functions on knots Statistics of linear families of smooth functions on knots

2010-06-07

arxiv.org/abs/1006.1267v2

Mathematics

Differential Geometry

13 pages, 2 figures

Scientific paper

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.

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