Statistics of linear families of smooth functions on knots

TL;DR

This paper derives formulas for the expected number of critical points of smooth functions on knots, linking geometric invariants to probabilistic properties, with explicit results for specific polynomial classes.

Contribution

It provides a new integral-geometric framework to compute expected critical points of functions on knots, connecting curvature and polynomial restrictions.

Findings

Expected critical points for random homogeneous polynomials: 2√(3d-2).

Expected critical points for trigonometric polynomials: approximately 1.549d.

Expressed the expected number in terms of total curvature and geometric invariants.

Abstract

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.