Geometric Constraints in Link Isotopy

2025/06/04 by José Ayala, Ayala, José
Mathematics · #Class (philosophy) #Curvature #Disjoint sets #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Isotopy #Link (geometry) #Linking number #Unknot #Writhe

paper · pdf · doi:10.48550/arxiv.2506.04442

openalex publication_date 2025/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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 gordian unknots: embedded tubes that are topologically trivial but geometrically locked, confirming a long-standing conjecture. These arise within the space U₁ of thin unknots in ℝ³, and persist across a stratified family \ U_τ\_{τ∈ [0,2]}, where τ 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.

Citations

• Thin Gordian Unlinks
• On the minimum ropelength of knots and links
• Gordian Unknots
• On Curves of Minimal Length with a Constraint on Average Curvature, and with Prescribed Initial and Terminal Positions and Tangents

Related

• Thin Gordian Unlinks
• A recipe for exotic 2-links in closed 4-manifolds whose components are topological unknots
• Is every knot isotopic to the unknot?
• Geometric Knot Spaces and Polygonal Isotopy
• Gordian split links in the Gehring ropelength problem