17 problems

Let $q>1$ and let $(L,\varphi)$ be an $F_m$-link concordance class in $C_{2q+1}(F_m)$. For each positive integer $k$, let $R_k(F_m)$ denote the space of $k$-dimensional unitary rep…

Let $n\geq 1$. Consider the ribbon link and its self-mutant described above, with all minimal linkings of weight at most $2n+7$ vanishing except for the specific $(2n+7)$-linking,…

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 $W_4$ 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.…

Feller–Lewark conjecture. For every link $L$ and every $g$, condition (1) is equivalent to condition (3); that is, $L$ bounds a $\mathbb Z$-surface of genus $g$ if and only if it a…

Let $\mathcal{T}^{\mathrm{tw}}_n$ be the group of order-$n$ twisted Whitney-tower intersection trees, let $\mathsf{W}^{\mathrm{tw}}_n$ be the corresponding graded quotient of links…

Conway coefficient sign conjecture. The conclusion of the two-component obstruction holds whenever $L\in\mathcal{P}_0$: in general, the sign $(-1)^k$ is replaced by $(-1)^{s(k,m)}$…

Boundary-link fusion conjecture. The class of links concordant to fusions of boundary links is equal to the class of links all of whose Milnor $\overline{\mu}$-invariants vanish.

A link is height $3.5$ Whitney tower concordant to the Hopf link if it is related to the Hopf link by a height $3.5$ Whitney tower concordance. Friedl and Powell constructed a Cass…

Let $k$ be a positive integer, and consider the horizontal Whitney-tower sequence in which ${\sf K}^{\mu}_{4k-3}$ is the middle group. Let ${\sf L}'_{2k}$ denote the relevant degre…

Let $K$ be a knot in $S^3$, and let $P(K)$ denote its $(2,0)$ cable, regarded as a parallel two-component link. A link is smoothly (respectively, topologically) concordant to anoth…

Let $L$ be a link in a homotopy $\mathbb R^3\times[0,1]$, and let its $s$-filtration be the filtration on the Lee homology associated to $L$. A concordance between links is an embe…

Let $L$ be a link in $S^3$, and let the obstructions of Theorem be those defined in the paper for detecting whether $L$ is concordant to the Hopf link. Whitney tower/Grope obstruct…

For each $k$, let $\mathcal{T}^{\infty}_{4k-2}$ be the twisted tree group and let $\text{\sf W}^{\infty}_{4k-2}$ be the corresponding twisted graded group of link concordance class…

Let $\mathcal{T}_n$ be the tree group of order $n$, let $\text{\sf W}_n$ be the corresponding graded group of link concordance classes, and set…

Vanishing Milnor invariants conjecture. Every link with vanishing Milnor $\bar\mu$-invariants is an $\widehat F$-link.