23 problems

Let $K$ be a nontrivial knot, let $N$ be a positive integer, and let $n$ satisfy $\left|n-\frac{N}{2}\right|<N^\delta$ for some $\frac{1}{2}<\delta<\frac{2}{3}$. Write…

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…

Let $K$ be a knot and $P$ a pattern whose pattern-defining link $P'$ has a well-defined GM series. Denote the satellite by $C_P(K)$. Satellite Formula. The GM series of $C_P(K)$ is…

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, and let $D_1$ and $D_2$ be slice disks for $K$ in $B^4$. Their positive Whitehead doubles are the slice disks obtained by applying the positive Whitehead satelli…

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 $P$ be a pattern with winding number, the algebraic intersection number of the pattern with a meridional disk of the solid torus, a nonzero even integer. For a knot $K$, write…

Let $P$ be a pattern knot in $S^1\times D^2$, let $K$ be a companion knot in $S^3$, and let $U$ denote the unknot in $S^3$. Satellite rank inequality. There is an inequality … This…

Winding-number lower-bound conjecture.

Consecutive-torus-knot conjecture. For every integer $n\geq 2$, if

Whitehead-double torus-knot conjecture. If $n$ is a positive integer and $K$ is the mirror of

Let $C\in\mathcal{G}_{\mathbb{Z}}$ have strongly detected boundary slopes $\mathcal{DS}_{C}$, and let $K=\operatorname{Sat}(P,C,f)$ be a satellite knot whose embedded pattern knot…

Lorenz satellite conjecture. A Lorenz knot that is a satellite has companion a Lorenz knot. Its pattern, when embedded in an unknotted solid torus in $S^3$, is equivalent to a Lore…

Let $P$ be a pattern, meaning a knot in a solid torus, and let $K$ be a knot in $S^3$. Write $P(K)$ for the satellite knot and $P(U)$ for the satellite of the unknot, and let…

Let a satellite knot be a knot whose exterior contains an incompressible torus. Let $\mathcal{Q}$ denote the class of quasi-alternating links, defined as the smallest set containin…

Let $K$ be a satellite knot and let $P$ be its companion. Let $\operatorname{cr}(X)$ denote the crossing number of a knot $X$. Satellite-knot crossing-number conjecture. The crossi…

Given a pattern knot $P$ and a companion knot $C$, both in the three-sphere, let $P(C)$ denote the resulting satellite knot. Hom–Lidman–Vafaee conjecture. If $P(C)$ is an L-space k…

Let $P$ be a pattern in the solid torus, let $K$ be a knot of genus $g$, let $U$ denote the unknot, and write $P(U,-n)$ for the pattern knot after $-n$ full twists. Satellite L-spa…

Let $P$ be a pattern in the solid torus, let $K$ be a knot, let $U$ denote the unknot, and write $P(K)$ and $P(U)$ for the corresponding satellite knots. An L-space knot is a knot…

Let $P$ be a pattern in the solid torus and let $K$ be a knot, with satellite knot $P(K)$. An L-space knot is a knot admitting a positive Dehn surgery yielding an L-space. Satellit…

Let $K$ be a satellite knot with companion $J$ and pattern $\hat{K}$ having wrapping number $n$. The width of a knot, denoted by $w(\cdot)$, is the knot width in the sense of Gabai…

Let $K$ be a Lorenz knot that is a satellite knot. Then the satellite Lorenz knot conjecture. $K$ is a satellite of a torus knot lying on a Lorenz knot, according to the cabling pa…