Elastic BB knots are multifold circles
Abstract
We study elastic knots obtained by minimizing the bending energy of a unit loop together with a small multiple of ropelength. We assume a generalized Fáry-Milnor inequality on the C1-boundary of a knot class: every H2-limit curve in the C1-closure has total curvature at least 2π times the bridge index. Under this assumption we prove that every elastic knot in a one-component BB knot class, that is, a class whose bridge and braid indices coincide, is the round circle covered precisely that many times. The proof does not differentiate the nonsmooth ropelength functional. A minimal closed braid is contracted in the fibres of a solid torus, producing a family of comparison curves with quadratic bending-energy excess and thickness of linear order. This yields a lower thickness bound for the regularized minimizers. Variations on an intermediate scale then give free stationarity of the limiting curve for the normalized bending energy. Constant curvature and free stationarity reduce the classification to a linear distributional ordinary differential equation.
Related
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- Elastic knots: compactness, contact, and variational structure
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BibTeX
@misc{reiter2026elastic,
title = {{Elastic BB knots are multifold circles}},
author = {Philipp Reiter and Heiko von der Mosel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22575},
eprint = {2609.22575},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22575},
}
RIS
TY - UNPB
TI - Elastic BB knots are multifold circles
AU - Philipp Reiter
AU - Heiko von der Mosel
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22575
UR - https://arxiv.org/abs/2609.22575
AB - We study elastic knots obtained by minimizing the bending energy of a unit loop together with a small multiple of ropelength. We assume a generalized Fáry-Milnor inequality on the C^1-boundary of a knot class: every H^2-limit curve in the C^1-closure has total curvature at least 2π times the bridge index. Under this assumption we prove that every elastic knot in a one-component BB knot class, that is, a class whose bridge and braid indices coincide, is the round circle covered precisely that many times. The proof does not differentiate the nonsmooth ropelength functional. A minimal closed braid is contracted in the fibres of a solid torus, producing a family of comparison curves with quadratic bending-energy excess and thickness of linear order. This yields a lower thickness bound for the regularized minimizers. Variations on an intermediate scale then give free stationarity of the limiting curve for the normalized bending energy. Constant curvature and free stationarity reduce the classification to a linear distributional ordinary differential equation.
ER -