Geometric Constraints in Link Isotopy

TL;DR

This paper demonstrates the existence of geometrically locked, topologically trivial unknots with prescribed physical constraints, revealing a complex stratified structure of isotopy classes due to geometric thresholds.

Contribution

It proves the existence of gordian unknots that are topologically trivial but geometrically locked, confirming a long-standing conjecture about their structure.

Findings

Existence of gordian unknots with prescribed constraints

Disconnected isotopy classes due to geometric thresholds

Stratified hierarchy of geometric entanglement

Abstract

We prove the existence of families of distinct isotopy classes of physical unknots through the key concept of parametrised thickness. These unknots have prescribed length, tube thickness, a uniform bound on curvature, and cannot be disentangled into a thickened round circle by an isotopy that preserves these constraints throughout. In particular, we establish the existence of \emph{gordian unknots}: embedded tubes that are topologically trivial but geometrically locked, confirming a long-standing conjecture. These arise within the space $\mathcal{U}_1$ of thin unknots in $\mathbb{R}^3$, and persist across a stratified family $\{ \mathcal{U}_\tau \}_{\tau \in [0,2]}$, where $\tau$ denotes the tube diameter, or thickness. The constraints on curvature and self-distance fragment the isotopy class of the unknot into infinitely many disconnected components, revealing a stratified structure governed by geometric thresholds. This unveils a rich hierarchy of geometric entanglement within topologically trivial configurations.