Let K be a tame knot class, let operatornamebra K denote its braid index, and let operatornamebri K denote its bridge index. An a-times covered circle is a circle traversed a times, and an elastic knot is a representative minimizing the bending energy within its knot class. Circular elastic knot conjecture. If operatornamebra K=operatornamebri K=a, then the a-times covered circle is the unique elastic knot for the tame knot class K. This extends the known comparison and lower-bound argument from the trefoil to knot classes whose braid and bridge indices coincide, and was previously stated by Gallotti and Pierre-Louis. The general assertion remains open in the supplied source.

Circular elastic knot conjecture

Let ${\mathcal K}$ be a tame knot class, let $\operatorname{bra}{\mathcal K}$ denote its braid index, and let $\operatorname{bri}{\mathcal K}$ denote its bridge index. An $a$ -times covered circle is a circle traversed $a$ times, and an elastic knot is a representative minimizing the bending energy within its knot class. Circular elastic knot conjecture. If

\operatorname{bra}{\mathcal K}=\operatorname{bri}{\mathcal K}=a,

then the $a$ -times covered circle is the unique elastic knot for the tame knot class ${\mathcal K}$ . This extends the known comparison and lower-bound argument from the trefoil to knot classes whose braid and bridge indices coincide, and was previously stated by Gallotti and Pierre-Louis. The general assertion remains open in the supplied source.

The conjecture predicts that whenever a knot class has equal braid and bridge indices, its unique minimum-bending representative is the corresponding multiply covered circle. It was predicted by Gallotti and Pierre-Louis in 2007 and formulated by Gerlach, Reiter, and von der Mosel in 2017.

Known results

The twice-covered circle was proved for $2$ -bridge knot classes, including the relevant $(2,b)$ -torus knots; the broader assertion remained conjectural in 2021.
The earlier trefoil result established uniqueness for the elastic trefoil and for $(2,b)$ -torus knots with odd $b$ and $|b|\geq 3$ .
The 2021 analysis showed that equal braid and bridge indices force equality in the bending-energy bounds and constant curvature, but not by itself a multiply covered circle.

2026 preprint claim

Tatsuya Miura’s Milnor’s inequality and circular elastic knots claims that extending Milnor’s inequality to the $C^1$ -closure of a knot class, combined with work of Reiter and von der Mosel, proves uniqueness of the $a$ -fold covered circle when $\operatorname{bri}(K)=\operatorname{bra}(K)=a$ . The preprint presents this as resolving the conjecture, but no independent verification or referee report was found.

Current status (as of September 2026): The general conjecture is claimed solved by Miura’s unrefereed preprint, while the claim remains unverified; the earlier $2$ -bridge and trefoil cases are established.

Circular elastic knot conjecture

Primary source

Progress summary

Solutions 0