Elastic knots: compactness, contact, and variational structure

Abstract: We study bending-minimizing sequences of closed curves in a fixed knot class and the self-contact of their limits. A positive matrix-valued measure records a possible loss of curvature energy. Ambient variations, localized on arbitrary coincidence sets, force this measure to vanish and give strong convergence in $H<sup>2$. We then derive the elastic equation on isolated strands and overlap bundles, describe the distribution that remains at contact, and prove balance and rigidity statements under their precise additional hypotheses. Equality in the bridge bound gives constant curvature. Combining strong compactness with quantitative competitors gives an alternative proof that elastic limits in classes with equal bridge and braid indices are covered circles. An independent recovery theorem identifies the strong $H<sup>2$ closure with the regular $H<sup>2$ part of the $C<sup>1$ closure and also implies strong compactness. Finally, a whole-line contact model illustrates which estimates are needed for finer shape selection.