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

TL;DR

This paper establishes lower bounds on the number of normals emanating from points to generic closed curves in three-dimensional space, using Morse theory and focal surface analysis.

Contribution

It introduces new bounds for the number of normals to closed curves in 3, extending previous understanding with Morse theory techniques.

Findings

At least 6 normals for smooth curves

At least 8 normals for piecewise linear curves

At least 10 normals if the curve is knotted

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.