Stability of knot equivalence at low regularity, and symmetric critical knots for the M\"obius energy

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 M\"obius 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 M\"obius energy.