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 H2. 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 H2 closure with the regular H2 part of the C1 closure and also implies strong compactness. Finally, a whole-line contact model illustrates which estimates are needed for finer shape selection.
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BibTeX
@misc{reiter2026elastic,
title = {{Elastic knots: compactness, contact, and variational structure}},
author = {Philipp Reiter and Heiko von der Mosel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26823},
eprint = {2610.26823},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26823},
}
RIS
TY - UNPB
TI - Elastic knots: compactness, contact, and variational structure
AU - Philipp Reiter
AU - Heiko von der Mosel
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26823
UR - https://arxiv.org/abs/2610.26823
AB - 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^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^2 closure with the regular H^2 part of the C^1 closure and also implies strong compactness. Finally, a whole-line contact model illustrates which estimates are needed for finer shape selection.
ER -