42 problems

Let $K\subset S^3=\partial D^4$ be a knot. It is slice if it bounds a smoothly, properly embedded disk in $D^4$; it is ribbon if it bounds a disk whose only singularities are ribbo…

Let $K$ be a slice knot in $S^3$, and let $F$ be a genus one Seifert surface for $K$. A slice derivative for $F$ is a simple closed curve on $F$ that is homologically nontrivial, h…

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…

The preceding construction concerns complementary slice discs and the transitivity of double $L$-groups. Product-structure conjecture. The techniques of double $L$-theory could be…

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.

Knot Floer homology conjecture. The group $\widehat{HFK}(J)$ is supported nontrivially in exactly $g$ diagonals, shifted up in Maslov grading from $K$. Hence the corresponding exot…

Let $C=11n_{34}$ denote the Conway knot. It has Alexander polynomial $1$ and is topologically slice but not smoothly slice. Conway knot ribbon minimality conjecture. The Conway kno…

The knot $6_1$ is the fourth twist knot, also denoted $T_4$. A slice disk is a disk bounded by the knot in the relevant four-dimensional setting, and two disks are considered equiv…

Slice-depth inequality conjecture. There exist $2$-knots whose slice depth is greater than their unknotting number.

Let $\includegraphics{mm.pdf}_{\ell}$ denote the $\ell$th knot in the Manolescu–Marengon family, and let $\operatorname{HFK}^-(\includegraphics{mm.pdf}_{\ell})$ be its minus knot F…

Finite slice-disk conjecture. The Stevedore knot $\mathbf{6_1}$ and the knots $\#^k\mathbf{9_{46}}$ have finite $\mathcal{D}(K)$; consequently, they provide examples for which the…

Let $K\subseteq S^3$ be a knot, and let $Wh^\pm(K)$ denote its positive or negative clasped, untwisted Whitehead double. Hedden's Whitehead-doubling conjecture. The knot…

Let $K$ be a knot in $S^3$. Its positive Whitehead double is the satellite knot obtained using the positive Whitehead pattern. Whitehead double sliceness conjecture. The knot $K$ i…

Let $E_{2,1}$ denote the $(2,1)$-cable of the figure eight knot. The topological sliceness question. Is $E_{2,1}$ topologically slice? This question is motivated by the fact that…

Let $K$ be a knot, let $p$ and $q$ be relatively prime integers with $|p|>1$, and let $K_{p,q}$ denote the $(p,q)$-cable of $K$. Let $\mathcal{C}$ denote the smooth knot concordanc…

A ribbon knot is a knot that bounds a ribbon disk in the $3$-sphere; a slice knot is a knot that bounds a smoothly embedded disk in the $4$-ball. The Alexander polynomial of a slic…

Let $K\subset S^3$ be a knot. A slice derivative is a derivative link $L$ for $K$ that bounds a collection of pairwise disjoint smooth disks in $B^4$. The stable Kauffman conjectur…

Odd pretzel strong double-slicing conjecture. The only strongly doubly slice odd pretzel links are of the form

Weak–strong double-slicing conjecture. There exists a link with doubly slice components that is weakly doubly slice but not strongly doubly slice.

Let $K_n$ be the family of knots considered in the paper, and set … A $G$-homotopy ribbon disc is a homotopy ribbon slice disc whose exterior has fundamental group $G$, considered…

Let $K_n$ be the family of knots considered in the paper, and set … A $G$-homotopy ribbon disc is a homotopy ribbon slice disc whose exterior has fundamental group $G$, considered…

Topological ribbon-slice conjecture. Every slice knot is homotopy ribbon. This is an open question about whether every topologically slice knot admits a slice disc satisfying the h…

Let $K$ be a knot, and let a slice derivative mean a derivative of $K$ that is slice in the relevant category. Generalised Kauffman conjecture. If $K$ is a topologically (respectiv…

Slice–ribbon conjecture. Every slice knot bounds an immersed disk in $S^{3}$ with only ribbon singularities.