10 problems

Let $\zeta(u):S^1\sqcup S^1\to S^3$ be the standard link of linking number $u$, and let $\sigma\zeta(u)$ denote the link obtained from $\zeta(u)$ by the operation $\sigma$ using a…

Cosmetic crossing conjecture. Changing a nontrivial crossing in a link diagram must change the isotopy type of the link.

Let $L_1$ and $L_2$ be links in the three-sphere, with crossing number denoted by $c(\,3c\,)$ and connected sum denoted by $\#$. Crossing-number additivity conjecture. For any two…

Let $\mathcal{L}$ be an $m$-component link, and suppose there are sub-links $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ with $i$ and $(m-i)$ components, respectively. For colorings…

Let $C_1$, $C_2$, and $C_3$ be any three planar curves. Planar-curve conjecture. They can be used to form the Borromean rings provided that at least one of them is not a circle. Th…

Let three unknotted curves in $\R^3$ be given, and suppose that at least one of them is not a circle. Noncircular obstruction conjecture. It is possible to choose such three curves…

Let $K_1$, $K_2$, and $K_3$ be three unknotted simple closed curves in $\R^3$. Cook's conjecture. They can always be arranged to form the Borromean rings unless they are all circle…

Let $(X,0)\subset (\mathbb{C}^3,0)$ be a surface with $1$-dimensional singular locus. An equisingular deformation of an irreducible curve means the total space of a one-parameter f…

Let a link be reducible by 3-moves if it can be transformed into a trivial link by a finite sequence of 3-moves. The Chen link is the closure of the 5-string braid…

Birman's conjecture. Every link has a single conjugacy class of minimal braids representing it.