75 problems

A pattern $P\subset S^1\times D^2$ induces a map $[P]\colon\mathcal{C}\to\mathcal{C}$ on the knot concordance group, given by $[K]\mapsto[P(K)]$. This map is a homomorphism when…

Surjectivity conjecture. The homomorphism $\Sigma$ is surjective.

Let $K\subseteq S^3$ be a knot, and write $K\#K$ for its connected sum with itself. A knot is negative amphichiral if it is isotopic to $r\overline{K}$, where $r$ denotes reversal…

Let $K_{0}$ and $K_{1}$ be fibered knots in $S^{3}$ supporting the tight contact structure. Baker's conjecture. If $K_{0}$ and $K_{1}$ are concordant, then $K_{0}=K_{1}$. The conje…

Let $K$) be a knot, let $D(K)$ denote its positively-clasped Whitehead double, and let $[K]$ and $[D(K)]$ denote their elements in the knot concordance group $\mathcal{C}$. Kirby’s…

Let $P\subset S^1\times D^2$ be a pattern, and let its winding number be the algebraic intersection number of $P$ with a meridional disk of $S^1\times D^2$. A pattern is rank expan…

Let $K$ be a knot, let $\overline{X}$ be the infinite cyclic cover of its exterior, and consider the associated Seifert-form data over $\mathbb{Q}$. A knot is $\mathbb{Q}$-anisotro…

Let $K$ and $K'$ be knots, and let $S^3_0(K)$ and $S^3_0(K')$ denote their 0-surgeries. Akbulut–Kirby conjecture. If $K$ and $K'$ have homeomorphic 0-surgeries, then $K$ and $K'$ a…

A knot $K$ is topologically slice if it bounds a topological slice disk $\Delta$ in $B^4$. It is topologically homotopy ribbon if it bounds such a disk for which the inclusion-indu…

Large four-genus conjecture. The collection $\{\#^n K\}$ consists of $h$-solvable knots and contains knots with arbitrarily large smooth four-genera $g_4$.

Let $\mathcal{C}$ denote the smooth knot concordance group. The knot invariant $r(K)$ is defined for knots $K$ and takes values in $\mathbb{Q}$. Concordance homomorphism conjecture…

Owens' H-thin conjecture. For any H-thin knot $K$,

Let $\mathcal F_n$ and $\mathcal F_{n.5}$ denote the successive terms of the Cochran–Orr–Teichner filtration of the smooth knot concordance group, indexed by $n\in\mathbb N_0$. Coc…

Let $K$ be a knot, and let $W_0(K)$ denote its zero-twisted Whitehead double. Smooth-sliceness conjecture for zero-twisted Whitehead doubles. The knot $K$ is smoothly slice if and…

A knot is ribbon minimal if it is minimal under ribbon concordance, and a link is ribbon concordance minimal if it is minimal under ribbon concordance. Component-separation conject…

Let $K$ be a knot in $S^3$, and let $\mu_K$ be a meridian of $K$. Define the two-component link … A link is ribbon concordance minimal if it has no strictly smaller link under ribb…

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…

Let $W_4$ denote the 4-twisted Whitehead link, a two-component link formed from the fourth twist knot and an unknot. Ribbon minimality conjecture. The link $W_4$ is ribbon minimal.…

Let $M\neq S^3$ be a closed orientable $3$-manifold, let $[S^1,M]$ denote the set of free homotopy classes of loops in $M$, and let $\mathcal{C}^M_x$ be the set of concordance clas…

Let $K$ and $K'$ be knots in $S^3$, and let $S^3_0(K)$ denote the $3$-manifold obtained by zero surgery on $K$. A knot is slice if it bounds a smoothly embedded disk in $B^4$. Slic…

For a Turk’s head knot $Th(p,q)$, let $g_4$ denote slice genus, $u$ unknotting number, and $g_3$ Seifert genus. Turk’s head genus conjecture. … and equality…

Satellite-operator dichotomy conjecture. Every satellite operator $P\colon \mathcal{C}\longrightarrow \mathcal{C}$ is constant or has image generating an infinite-rank subgroup of…

Let $\mathcal{C}^\mathrm{diff}$ and $\mathcal{C}^\mathrm{top}$ denote the smooth and topological knot concordance groups. A set has a fractal structure if it admits self-similariti…

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…