147 problems

Let $K$ be a knot, let $s(K)$ denote Rasmussen's invariant from Lee homology, and let $\tau(K)$ denote Ozsváth–Szabó's invariant from Heegaard Floer homology. Rasmussen's conjectur…

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…

A tight fibered knot is a fibered knot supporting the tight contact structure. A knot is prime if it cannot be expressed as a nontrivial connected sum. The smooth knot concordance…

Given a knot $K$, let the smooth non-orientable four-genus be … with $\gamma_4(K)=0$ when $K$ is slice. For a torus knot $T_{p,q}$, let $\vartheta(T_{p,q})$ denote the number of pi…

Let $K$ and $K'$ be knots in $S^3$. Their zero surgeries are the $3$-manifolds $S^3_0(K)$ and $S^3_0(K')$, respectively, and the knots are concordant when they cobound a smooth ann…

Hedden–Pinzón-Caicedo conjecture. The only homomorphisms induced by satellite operations are the following:

An algebraic knot is a knot arising from an algebraic plane curve singularity; these knots are, in particular, iterated cables of torus knots. The smooth knot concordance group is…

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…

A knot is called positive if it admits a positive diagram. A knot is $\mathbb{Q}$-anisotropic when its infinite-cyclic-cover cohomology has no nontrivial deck-transformation-invari…

Let $\mathcal{C}$ be the classical knot concordance group. An element of $\mathcal{C}$ is 2-torsion if it has order two, and a knot is negative amphichiral if it is isotopic to its…

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…

A satellite operation is an operation on knots obtained by placing a pattern knot inside a solid torus and tying that solid torus into a companion knot. The Cochran–Orr–Teichner fi…

Signature-detection conjecture. In general, signature invariants cannot detect elements in the kernel of $\phi_1$.

The Ozsváth–Szabó invariant $\tau$ and Rasmussen invariant $s$ are knot concordance invariants. For integers $p,q$ and $t$, let $D_+(T_{p,q},t)$ denote the $t$-twisted positive dou…

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

Kernel-structure conjecture.

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…

Higher-dimensional algebraic four-genus conjecture. In higher dimensions, this inequality should be an equality:

For a knot $K$, let $P_+$ and $P_-$ denote the $(2,1)$- and $(2,-1)$-cable patterns, respectively, and let $P$ be any pattern. Let $s_p$ denote the Rasmussen invariant over a field…

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…

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…

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…