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Stability of knot equivalence at low regularity, and symmetric critical knots for the Möbius energy

Simon Blatt, Alexandra Gilsbach, Philipp Reiter, Heiko von der Mosel

TL;DR

The paper develops a localized distortion framework to compare knot images and deduce isomorphisms between knot complements under low regularity, enabling stability results for knot equivalence in Lipschitz and fractional Sobolev spaces. It introduces distance-flow techniques and harmonic substitution to handle low-regularity minimizing sequences, and proves a weak fractional compactness theorem for symmetric $W^{3/2,2}$ knots. These tools are then applied to the Möbius energy, establishing the existence of symmetric minimizing knots within given knot classes that are real analytic and, in particular, yielding at least two symmetric critical torus knots per class. The work unifies geometric, topological, and variational methods to address knot stability at low regularity and symmetry-constrained criticality of scale-invariant energies.

Abstract

We present sufficient criteria for the equivalence of tame knots at low regularity. To this end, we introduce a localized version of Gromov's distortion for any closed path-connected subset of $\R^n$. If two such sets have local Gromov distortion below a universal dimension-dependent constant $g_n$ at some scale, and if their Hausdorff-distance is less than one quarter of that scale, we can show that the fundamental groups of their complements are isomorphic. In addition, we construct this isomorphism so that it restricts to the corresponding peripheral subgroups as an isomorphism as well. Applied to the images of one-dimensional knots it follows that two knots are equivalent if their Hausdorff-distance is bounded in terms of the scale under which their local Gromov distortion is controlled. From that we deduce novel stability results for knot equivalence in the Lipschitz category, and in the setting of fractional Sobolev regularity below $C^1$. Moreover, we prove a compactness theorem of knot equivalence classes with respect to weak $W^{3/2,2}$-convergence. As an application we show that the Möbius energy introduced by O'Hara~\cite{ohara_1991a} can be minimized within arbitrary prime knot classes under a symmetry constraint, and that these minimizers are in fact critical points and therefore smooth and even real analytic. In particular, in every torus knot class there are at least two distinct critical knots for the Möbius energy.

Stability of knot equivalence at low regularity, and symmetric critical knots for the Möbius energy

TL;DR

The paper develops a localized distortion framework to compare knot images and deduce isomorphisms between knot complements under low regularity, enabling stability results for knot equivalence in Lipschitz and fractional Sobolev spaces. It introduces distance-flow techniques and harmonic substitution to handle low-regularity minimizing sequences, and proves a weak fractional compactness theorem for symmetric knots. These tools are then applied to the Möbius energy, establishing the existence of symmetric minimizing knots within given knot classes that are real analytic and, in particular, yielding at least two symmetric critical torus knots per class. The work unifies geometric, topological, and variational methods to address knot stability at low regularity and symmetry-constrained criticality of scale-invariant energies.

Abstract

We present sufficient criteria for the equivalence of tame knots at low regularity. To this end, we introduce a localized version of Gromov's distortion for any closed path-connected subset of . If two such sets have local Gromov distortion below a universal dimension-dependent constant at some scale, and if their Hausdorff-distance is less than one quarter of that scale, we can show that the fundamental groups of their complements are isomorphic. In addition, we construct this isomorphism so that it restricts to the corresponding peripheral subgroups as an isomorphism as well. Applied to the images of one-dimensional knots it follows that two knots are equivalent if their Hausdorff-distance is bounded in terms of the scale under which their local Gromov distortion is controlled. From that we deduce novel stability results for knot equivalence in the Lipschitz category, and in the setting of fractional Sobolev regularity below . Moreover, we prove a compactness theorem of knot equivalence classes with respect to weak -convergence. As an application we show that the Möbius energy introduced by O'Hara~\cite{ohara_1991a} can be minimized within arbitrary prime knot classes under a symmetry constraint, and that these minimizers are in fact critical points and therefore smooth and even real analytic. In particular, in every torus knot class there are at least two distinct critical knots for the Möbius energy.

Paper Structure

This paper contains 13 sections, 26 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Isomorphic fundamental groups via localized distortion
  4. Stability of knot equivalence
  5. Symmetric critical points for scale-invariant knot energies
  6. Strategy and outline of the paper
  7. Sets with local distortion below a quarter circle
  8. Pseudo-gradient flow for the distance function
  9. Isomorphisms of fundamental groups
  10. Stability of knot equivalence
  11. Harmonic substitution
  12. Weak fractional compactness
  13. Symmetric critical knots

Key Result

Theorem A

For any immersed embedded curve $\gamma\in C^1(\mathbb{R}/\mathbb{Z},\mathbb{R}^3)$ there is some number $\epsilon_\gamma>0$ such that all curves $\eta\in C^1(\mathbb{R}/\mathbb{Z},\mathbb{R}^3)$ with $\|\gamma-\eta\|_{C^1} < \epsilon_\gamma$ are embedded and ambiently isotopic to $\gamma$.

Theorems & Definitions (42)

  • Theorem A: $C^1$-isotopy stability diao-etal_1999, reiter_2005, denne-sullivan_2008
  • Theorem B: Knot equivalence via knot complements waldhausen_1968gordon-luecke_1989
  • Definition 1.1
  • Theorem 1.2: Isomorphic fundamental groups
  • Corollary 1.3: Knot equivalence via local distortion
  • Corollary 1.4: Lipschitz stability of knot equivalence
  • Corollary 1.5: Fractional stability of knot equivalence
  • Theorem 1.6: Symmetric critical prime knots
  • Corollary 1.7: Symmetric critical prime knots in ambient isotopy classes
  • Lemma 2.1: Diameter bound on set of directions
  • ...and 32 more