18 problems

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.

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…

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…

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 $\mathcal{U}_{w,F}$ 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…

Fix a ribbon knot and consider ribbons spanning it. A ribbon 0-move, ribbon 1-move, finger move, and ribbon 3-move are the ribbon transformations described in the paper's figures;…

Let $K$ be a knot in $S^3$. A concordance is a cobordism of minimal intrinsic complexity, and it has minimal extrinsic complexity when it has no index $2$ Morse critical points. Sl…

For a knot $K$, let its ribbon genus be the minimum genus of a ribbon surface bounding it, and let $b(K)$ denote its band number. Fixed-genus band number conjecture. For every genu…

Let $\mathcal{B}(R,S,T)$ be the ribbon-knot family considered in the paper, with parameters $R,S,T$. A ribbon is respected when a transformation to a symmetric union preserves that…

Let $\mathcal{A}(S,T)$ be the ribbon-knot family considered in the paper, with parameters $S,T$, and let 12n268 denote the specified knot in the determinant-$9$ table. A symmetric…