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

Abstract

We prove the following results: (1) For every generic closed smooth curve in ${\mathbb{R}}^3$, {whose torsion has a constant sign}, there is a point with at least $6$ emanating normals to the curve. If the curve is knotted, and torsion has a constant sign, there is a point with at least $8$ emanating normals. (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 {(no condition on the torsion)}. If the curve is knotted, then 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.