43 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…

Let $K$ be a satellite knot with companion $J$ and wrapping number $m$. The Zupan width conjecture. … The conjecture replaces the winding number by the always nonzero wrapping numb…

Let $K$ be a satellite Lorenz knot, with pattern $P$ and companion $C$ in the standard satellite construction. Lorenz satellite conjecture. The companion $C$ and pattern $P$ of eve…

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…

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…

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…

Let $p,q$ be positive integers with $\gcd(p,q)=1$, and let $K$ be the $(p,q)$-cable knot about a knot $C$ in $\mathcal{F}$. Cable crossing-number conjecture. The crossing number sh…

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…

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…

Satellite crossing number conjecture. The following inequalities are conjectured:

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…

Satellite knot crossing-number conjecture. One has

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 $K$ be a knot with $\varepsilon(K)=\pm1$, and let $P$ be a pattern with wrapping number two and winding number $\pm2$. Let $\vartheta_\tau(P)$ be the corresponding satellite in…

Let $K$ be a $\vartheta_c$-rational knot, and let $P$ be any pattern, with arbitrary winding and wrapping number. Arbitrary-pattern Rasmussen satellite conjecture. … This is propos…

A knot $K$ is $\vartheta_c$-rational when its associated tangle invariant has the form of the invariant of a rational tangle. Vanishing conjecture for $\vartheta_c$-rational knots.…

Let $K\subset S^3$ be a knot, let $c$ be either $0$ or a prime, and let $P$ be a pattern with wrapping number two and winding number zero. Write $P_t(K)$ for the $t$-twisted satell…

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

Lewark–Zibrowius conjecture. For every prime $p$ and every prime $q\ne p$,

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