7 problems

Let $L$ be a link, let $R^p(L)$ be the scale-invariant energy defined in the paper, and let $R(L)$ denote ropelength, with thickness measuring the radius of the largest embedded tu…

Piecewise analyticity conjecture. Ropelength minimizers are piecewise analytic.

Diameter variation conjecture. For each ropelength level $\Lambda$, there should exist constants $C(\Lambda)>0$ and $\alpha>0$ such that

Non-degeneracy conjecture. The swept-area pseudometric is non-degenerate on each admissible component of $Y_{\Lambda}(K)$ modulo $\operatorname{Isom}^+(\mathbb{R}^3)$: zero swept-a…

Finite witness principle. For each $\mathcal I\in\mathfrak I$, each knot type $K$, and each generic projection direction $u$, there exists a finite witness scale

Geometric decomposition conjecture. Every length-minimising unknot in $\mathcal{U}_\tau$ for $\tau\in(0,2)$ admits a decomposition into finitely many elementary geometric component…

Let $L$ be a link that is balanced in the sense of the paper's balance criterion, and let a link be nicely balanced when its kink measure is represented by a kink tension function…