Miura's second-energy conjecture for the elastic propeller

From papers

For a tame knot class KK, let K\mathcal{K} be the W2,2W^{2,2}-closure of the class of unit-length, arclength-parametrized closed curves in KK:

K:={γW2,2(R/Z;R3)γ1, γK}.\mathcal{K}:=\overline{\{\gamma\in W^{2,2}(\mathbf{R}/\mathbf{Z};\mathbf{R}^3)\mid |\gamma'|\equiv1,\ \gamma\in K\}}.

A stable elastic knot for KK is a curve γK\gamma\in\mathcal{K} that locally minimizes the bending energy BB on a connected component of the knot class, and it is strictly stable if the minimum is strict up to isometry. Elastic propeller conjecture. The elastic propeller is a strictly stable elastic knot for the unknot class. Moreover, among all stable elastic knots for the unknot class, it has the second smallest energy. The circle is the least-energy stable elastic unknot, so the conjecture identifies the elastic propeller as the next energy level. The source attributes this conjecture to Miura; its resolution is not specified.

Progress summary

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Sources & referencesView supporting material

Primary source

Tatsuya Miura, “Elastic curves and self-intersections”, arXiv:2408.03020 (2025).

Solutions 0

No solutions have been posted yet.