11 problems

For $n>2$, consider the tight contact structures $\eta^n_{i,j}$ on $-\Sigma(2,3,6n-1)$ with $0\leq i\leq n-2$, $|j|\leq n-i-2$, and $j\equiv n-i\pmod{2}$, and the tight contact str…

Planarity conjecture. No tight, positive contact structure on a Brieskorn homology sphere is supported by a planar open book decomposition.

Kollár's conjecture. No Brieskorn sphere $\Sigma(a_1,\ldots,a_n)$ with $n\geq 4$ is the boundary of a smooth integer homology ball.

Gompf's conjecture. No nontrivial Brieskorn sphere, with either orientation, arises as the boundary of a Stein domain in $\mathbb{C}^2$.

Let $\Sigma(p,q,r)$ be a non-trivial Brieskorn homology sphere, let $n$ be a natural number, and let $K$ be a regular Lagrangian slice knot. Lagrangian-surgery conjecture. No such…

Let $\Sigma(p,q,r)$ be a Brieskorn homology sphere with its usual orientation, and let $-\Sigma(p,q,r)$ denote the same manifold with the opposite orientation. Let $K_{p,k}$ denote…

Let $\Sigma(p,q,r)$ be a Brieskorn homology sphere, and let $K_{p,k}$ denote a parameterized lens space knot. Existence conjecture. Every Brieskorn homology sphere contains a knot…

Let $K_{p,k}$ and $K_{p',k}$ denote the parameterized dual knots in the tables for $\Sigma(2,3,5)$ and $\Sigma(2,3,7)$, respectively, and let $g(K)$ denote the knot genus. Complete…

Let $K_{p',k}$ be any knot whose parameter $(p',k)$ occurs in the displayed table for lens space surgeries in $\Sigma(2,3,7)$, and let $g'$ be the table's value in its $g'$ column.…

Let $p>1$ be an odd integer, let $k$ be an integer satisfying $\gcd(k,2p)=1$ and $k\not\equiv\pm1\pmod{2p}$, and let $n$ be an integer. For a rational homology sphere $Y$, write…

Non-Stein-fillability conjecture. The contact structures $\eta_{i,j}^n$ are not Stein fillable if $i>0$.