15 problems

Expansion conjecture. The space $Y$ smoothly embeds in $S^4$ if and only if it is obtained by a possibly empty sequence of expansions from some $Y'$ of the displayed form, with

Let $Y=Y(e_0;r_1,r_2,\ldots,r_n)$ be a Seifert fibered space that is also a rational homology sphere, and let $\xi$ be a contact structure on $Y$. Seifert filling classification co…

The $\mathcal{QHB}$ conjecture. If $Y$ admits a symplectic rational homology ball filling, then

Gordon's conjecture. If $S^3_{p/q}(K)$ is a Seifert fibered space, then

Finiteness conjecture. Let $M$ be a compact oriented 3-manifold. Then $\mathcal{S}(M)$ is a finitely generated $\mathcal{S}(\partial M, \mathbb{Q}(A))$-module.

Integrality conjecture. Any Seifert-fibered surgery on a hyperbolic knot in $S^3$ is integral.

Let $K$ be a knot in $S^3$, let $p/q\in\mathbb{Q}$ be a surgery slope with $q\geq 2$, and write $S_{p/q}^3(K)$ for the manifold obtained by $p/q$-surgery on $K$. A Seifert fibred s…

Link-invariant algebraicity conjecture. All perturbative colored HOMFLY invariants of links in $M$ belong to $L$, and all perturbative colored Kauffman invariants of links in $M$ b…

Algebraicity conjecture. The extension $L$ can be chosen depending only on the ambient manifold and the Lie-algebra series $X\in\{A,BCD\}$, and, at least for fiber knots, the pertu…

Fix a $2$-orbifold $S^2(k,l,m)$ with $k,l,m>1$. Consider Dehn surgeries on hyperbolic knots in $S^3$ that produce Seifert fibred spaces having $S^2(k,l,m)$ as their base orbifold.…

Let $K$ be a hyperbolic knot in $S^3$. A surgery on $K$ is a Dehn surgery whose resulting manifold is a Nil Seifert fibred space. Nil surgery conjecture. If $K$ admits a surgery yi…

Seifert fibering conjecture. If non-integer surgery on $\kappa$ is Seifert fibred, then $\kappa$ is a torus knot or a cable of a torus knot.

Let $m$ and $n$ be positive integers, and consider the Seifert-fibered dihedral manifolds … A knot surgery is a Dehn surgery on a knot in the $3$-sphere, and the family is consider…

C. McA. Gordon's conjecture. Suppose that $M$ is a hyperbolic knot manifold $M$ and $\alpha, \beta$ are slopes on $\partial M$ such that $M(\alpha)$ is small Seifert and $M(\beta)$…

A non-flat non-orientable Seifert fibred space over $\mathbb{R}P^2$ or $\bar{D}$ with two exceptional fibres is a Seifert fibred 3-manifold in the stated class, where $\bar{D}$ den…