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arXiv · 2609.27643

Milnor's inequality and circular elastic knots

Tatsuya Miura
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Abstract

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.

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Tatsuya Miura. 2026-09-23. Milnor's inequality and circular elastic knots. https://arxiv.org/abs/2609.27643

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