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$h$-Principles for curves and knots of constant torsion

Mohammad Ghomi, Matteo Raffaelli

TL;DR

This work proves a $C^1$-dense $h$-principle for curves in $\mathbb{R}^3$ with constant torsion, ensuring knots exist in every isotopy class with $\widetilde{\tau}=c$ for some $c>0$. The authors reduce the problem to deformations of the tantrix on the sphere, then use a unique reparametrization to enforce $\widetilde{k}\widetilde{v}=c$, followed by a convex-integration/degree-theory framework to realize global homotopies while preserving prescribed tangencies. A local spherical-curve problem is solved and lifted to the global setting, with an explicit construction that controls the tantrix average via loop insertions and large torsion. The results extend analogous $h$-principles for constant curvature and provide insight into elastic rod knotting under geometric constraints, with potential implications for isotopy and geometric modeling of curves.

Abstract

We prove that curves of constant torsion satisfy the $C^1$-dense h-principle in the space of immersed curves in Euclidean space. In particular, there exists a knot of constant torsion in each isotopy class. Our methods, which involve convex integration and degree theory, quickly establish these results for curves of constant curvature as well.

$h$-Principles for curves and knots of constant torsion

TL;DR

This work proves a -dense -principle for curves in with constant torsion, ensuring knots exist in every isotopy class with for some . The authors reduce the problem to deformations of the tantrix on the sphere, then use a unique reparametrization to enforce , followed by a convex-integration/degree-theory framework to realize global homotopies while preserving prescribed tangencies. A local spherical-curve problem is solved and lifted to the global setting, with an explicit construction that controls the tantrix average via loop insertions and large torsion. The results extend analogous -principles for constant curvature and provide insight into elastic rod knotting under geometric constraints, with potential implications for isotopy and geometric modeling of curves.

Abstract

We prove that curves of constant torsion satisfy the -dense h-principle in the space of immersed curves in Euclidean space. In particular, there exists a knot of constant torsion in each isotopy class. Our methods, which involve convex integration and degree theory, quickly establish these results for curves of constant curvature as well.
Paper Structure (4 sections, 6 theorems, 10 equations, 1 figure)

This paper contains 4 sections, 6 theorems, 10 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Reparametrization of the Tantrix
  3. Reduction to Spherical Curves
  4. Controlling the Average

Key Result

Theorem 1.1

Let $f\in\textup{Imm}^{\alpha\geqslant 4}(\Gamma,\mathbf{R}^3)$ be a unit speed curve with $\kappa$, $\tau>0$, and $p_i\in\Gamma$ be a finite collection of points. Then for any $\varepsilon>0$ there exists a unit speed curve $\widetilde{f}\in\textup{Imm}^{\alpha-1}(\Gamma,\mathbf{R}^3)$ with $\widet

Figures (1)

  • Figure 1:

Theorems & Definitions (8)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 4.1
  • proof