2025/12/02 by Simon Blatt, Blatt, Simon, Alexandra Gilsbach +5
Mathematics · #49Q10 #57M05 #57M25 #57M60 #Bounded function #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #Equivalence (formal languages) #FOS: Mathematics #Fibered knot #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Knot (papermaking) #Knot invariant #Knot theory #Quantum invariant
paper · open access · doi:10.48550/arxiv.2512.02998
published in RWTH Publications (RWTH Aachen) (RWTH Aachen University)
openalex publication_date 2025/12/02 · openalex created_date 2025/12/04 · openalex updated_date 2026/08/05
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 \Rn. If two such sets have local Gromov distortion below a universal dimension-dependent constant gn 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 C1. Moreover, we prove a compactness theorem of knot equivalence classes with respect to weak W3/2,2-convergence. As an application we show that the Möbius energy introduced by O'Hara~\citeohara1991a 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.