Counting normals to closed curves in $\mathbb{R}^3$

Abstract

We prove the following results: (1) For every generic closed smooth curve in $\mathbb{R}^3$ there is a point with at least $6$ emanating normals to the curve. (2) For every generic closed piecewise linear curve in $\mathbb{R}^3$ there is a point with at least $8$ emanating normals to the curve. If the curve is knotted, there is a point with at least $10$ emanating normals. The proof is based on the Morse theory for the squared distance function and self intersections of the focal surface.