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Milnor's inequality and circular elastic knots
Abstract
We establish the extension of Milnor's inequality, relating total curvature with the bridge index, to the C1-closure of a knot class, without any restriction on the self-intersections of the limit curve. Together with recent work of Reiter--von der Mosel, this resolves the circular elastic knot conjecture, first predicted by Gallotti--Pierre-Louis in 2007 and then formulated as a mathematical conjecture by Gerlach--Reiter--von der Mosel in 2017. More precisely, if the bridge and braid indices of a tame knot class coincide, then the multiply covered circle is the unique elastic knot.
Related
BibTeX
@misc{miura2026milnors,
title = {{Milnor's inequality and circular elastic knots}},
author = {Tatsuya Miura},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27643},
eprint = {2609.27643},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27643},
}
RIS
TY - UNPB
TI - Milnor's inequality and circular elastic knots
AU - Tatsuya Miura
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27643
UR - https://arxiv.org/abs/2609.27643
AB - We establish the extension of Milnor's inequality, relating total curvature with the bridge index, to the C^1-closure of a knot class, without any restriction on the self-intersections of the limit curve. Together with recent work of Reiter--von der Mosel, this resolves the circular elastic knot conjecture, first predicted by Gallotti--Pierre-Louis in 2007 and then formulated as a mathematical conjecture by Gerlach--Reiter--von der Mosel in 2017. More precisely, if the bridge and braid indices of a tame knot class coincide, then the multiply covered circle is the unique elastic knot.
ER -