19 problems

Let $K$ be a knot. Its ropelength $R(K)$ is the minimum length among all representatives of the ambient isotopy class of $K$ with unit thickness, and let $Cr(K)$ denote its crossin…

For $m=1,2$, let $L^m(\tau)$ be the different critical configuration described for the generalized link $C_{0,2}^m(\tau)$, with $k=0$ and $l=2$. Minimizer conjecture. In both cases…

Let $L$ be a critical link, and let its curvature measure be the measure describing the curvature of $L$ along the link. Curvature-measure absolute continuity conjecture. The curva…

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…

Let $L$ be a link, and for $p\geq 1$ let $R^p(L)$ denote the scale-invariant $L^p$ energy defined by integrating inverse powers of the radius function $r(T_xL,y)$. An energy is bas…

Disjoint-disk conjecture. The intersection of the tube around $K$ with some $4\pi$ cone on $K$ contains $\operatorname{PC}(K)$ disjoint unit disks avoiding the cone point.

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…

A trefoil knot can be placed in configurations with 3-fold or 2-fold rotational symmetry. Moffatt's conjecture. The 3-fold symmetric and 2-fold symmetric configurations of the tref…

Let $K_1$ and $K_2$ be knots, let $K_1\# K_2$ denote their connect sum, and let $\operatorname{Wr}(K)$ denote the average writhing number of a tight configuration of $K$. Katritch…

A supercoiled helix is one of the curves obtained by numerically integrating the differential equation governing a kink-tension function in the kink-measure-only case. Embeddedness…

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…

Let $L$ be a link that is balanced by a strut measure and a kink measure. It is regularly balanced if there is an open subset $U\subset L$ such that the projected kink measure vani…

A link $L$ is a critical link if it is critical for length under the thickness constraint. Its kinks are regulated when the tangent has a regulated derivative on the relevant open…

A mirror-composite knot is a composite knot formed by joining a knot to its mirror image. The Gonzalez conjecture. Every mirror-composite knot has a critical configuration with a p…

Let $(\mathcal{V}_n)$ be a family of polygonal ropelength-critical configurations converging to a $C^{1,1}$ curve $V$. The polygonal approximation conjecture. The curve $V$ is rope…