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…

Let $K$ be a knot or link, let $c(K)$ denote its crossing number, and let $\mathrm{Rib}(K)$ denote its ribbonlength. For constants $c_1,c_2>0$ and exponents $\alpha,\beta$, Diao–Ku…

Let $[\gamma]$ be a knot type, let $U_C([\gamma])$ be the set of curves in that knot type whose distortion is at most $C$, and let $\delta([\gamma])$ be the distortion thickness of…

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…

Compression lower bound from supertrunk. There is a universal constant $c>0$ such that for every knot type $K$,

Stratified obstruction theory. For each $\tau\in(0,2]$, the space $\mathcal{U}_\tau$ contains countably infinite isotopy classes. As $\tau\to0$, geometric obstructions, including b…

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 $K$ be a ribbon knot, let $\operatorname{Rib}(K)$ denote its minimal ribbonlength, and let $c(K)$ denote its crossing number. Linear ribbonlength conjecture. The minimal ribbon…

Let $\Gamma_n$ denote the set of convex $n$-gons with each edge of length $1$, and let $E_n^\alpha$ be the energy associated with the increasing continuous function…

Let $[K]$ be a knot type, let $s_{sh}[K]$ denote its stick number in the simple hexagonal lattice, and let $c[K]$ denote its crossing number. Crossing-number bound conjecture. For…

Let $[K]$ be a knot type, let $s_{sh}[K]$ denote its stick number in the simple hexagonal lattice, and let $3_1$, $4_1$, $5_1$, and $5_2$ denote the indicated knot types. 13-stick…

Let $\gamma$ be a curve in the fractional Sobolev space $W^{\frac{3}{2},2}$, and let $E^1(\gamma)$ denote the first component in Ishizeki and Nagasawa's decomposition of the Möbius…

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…

Let $(p,q)$ be integers, and let a $(p,q)$ torus knot mean the corresponding isotopy class of knots. A critical point of $E^{(2)}_{\circ}$ is unstable if the energy has a descendin…

Let $[K_1]$ and $[K_2]$ be isotopy classes of knots, and let $[K_1\sharp K_2]$ be their composite isotopy class. For an isotopy class $[K]$, write $E^{(2)}_{\circ}([K])$ for the in…