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arXiv · 2602.06645

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

Gaiane Panina·Dirk Siersma
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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.

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Gaiane Panina, Dirk Siersma. 2026-09-20. Counting normals to closed curves in $\mathbb{R}^3$. https://arxiv.org/abs/2602.06645

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