34 problems

Determinant-square conjecture. In general, the maximum determinant of a prime symmetric ribbon knot $K$ with $r_s(K)=n$ is the square of the maximum determinant of a prime knot $J$…

Unbounded-gap conjecture. For every positive integer $n$, there exists a symmetric ribbon knot $K$ such that

A knot is topologically slice if it bounds a locally flat embedded disc in $D^4$, and it is homotopy ribbon if it bounds a disc whose complement has the corresponding ribbon-type f…

Ribbon-knot symmetric-union conjecture. Every ribbon knot is a symmetric union.

Casson–Gordon's conjecture. Every 2-bridge ribbon knot is contained in one of the three families of Casson and Gordon.

Vanishing Hom conjecture. For $i=1,2$,

Ribbon group asphericity conjecture. Every ribbon group $G$ is aspherical. In particular,

H-ribbon characterization conjecture. A knot $K$ is h-ribbon with group $G$ if and only if there exists an epimorphism $\varphi\colon \pi_1(M_K)\twoheadrightarrow G$ such that cond…

Satellite h-ribbon conjecture. Theorem gives new examples of knots which are h-ribbon but not smoothly slice.

Equivariant slice-ribbon conjecture. An equivariant slice knot is equivariant ribbon.

Ribbon-number-two conjecture. If $r(K)=2$, then

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…

Ribbon–SU conjecture. Every ribbon knot is an SU knot.

Let $p,q\in\mathbb{Z}$ be odd, let $n\in\mathbb{N}$, and let $K$ be the $(2n+1)$-stranded pretzel knot … For $0\leq g\leq n$, write $r_g(K)$ for the higher-genus ribbon number and…

Let $K$ be a knot, let $\overline{K}$ denote its mirror image, let $r(K)$ denote the minimum ribbon number of a ribbon disk for $K$, and let $c(K)$ denote the crossing number of…

Let $K_1$ and $K_2$ be ribbon knots, and let $r(K)$ denote the minimum ribbon number of a ribbon disk for $K$. Ribbon number additivity conjecture. If $K_1$ and $K_2$ are ribbon kn…

A knot $K$ is a symmetric union if it is obtained from the connected sum $J\#-J$ of a knot $J$ with its mirror $-J$ by replacing trivial 4-tangles intersecting the symmetry axis wi…

A ribbon knot is a knot that bounds a singular disk with only ribbon singularities, and a symmetric union is the type of knot presentation referred to in the conjecture. Lamm's con…

A knot is called slice if it is concordant to the unknot, and is called ribbon if it bounds a singular disk with only ribbon singularities. Slice-ribbon conjecture. Every slice kno…

A ribbon knot is a knot bounding an immersed ribbon disc in $B^4$, with ribbon singularities. Non-locality conjecture. There cannot exist a local proof of the modulo-$4$ ribbon-kno…

Let $R\subset S^3$ be a ribbon knot, and let $\mathcal{A}$ be either $\mathbb{F}_p$ or $\mathbb{Q}$. Write $\widehat{\operatorname{HFK}}(R)$ for hat knot Floer homology and…

Let $R\subset S^3$ be a ribbon knot, and let $\mathcal{A}$ be either $\mathbb{F}_p$ or $\mathbb{Q}$. Write $\widehat{\operatorname{HFK}}(R)$ for hat knot Floer homology and…

Let $n$ be a nonnegative integer, and let $\mathcal{U}_{w,F}$ be a folded ribbon unknot that is a topological annulus, with ribbon linking number…

Let $n\geq4$, and let $\mathcal{U}_{w,F}$ be a folded ribbon convex $n$-stick unknot. Let ${\operatorname{Rib}}$ denote folded ribbonlength. Regular polygon conjecture. The minimum…

Let $\mathcal{U}_{w,F}$ be a folded ribbon unknot that is a Möbius band, with ribbon linking number ${\operatorname{Lk}}(\mathcal{U}_{w,F})=\pm3$. Let ${\operatorname{Rib}}$ denote…