30 problems

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

Let $G$ be a Seifert $3$-manifold group. Its single equation problem asks whether a single equation over $G$ has a solution. Theorem B establishes decidability when the base orbifo…

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…

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 $K$ be a knot in $S^3$ such that some surgery on $K$ yields a Seifert fiber space. A knot $c\subset S^3$ is trivial if it is an unknot, and it is disjoint from $K$ when…

Integral Seifert fibered slope conjecture.

Let $K$ be a hyperbolic knot in the $3$-sphere $S^3$, and let $r\in\mathbb{Q}\cup\{\infty\}$ be a surgery slope. The resulting manifold $(K;r)$ is obtained by Dehn surgery on $K$ w…

Let $(K;\gamma)$ be a Seifert fibered surgery on a knot $K$ in $S^3$, where $(K;\gamma)$ denotes the $3$-manifold obtained by Dehn surgery on $K$ along the slope $\gamma$. Dean's p…

Let $w_{p,q,r,m}$ denote the words considered in the paper, and suppose the three types of Seifert-fibered elements are those described in Propositions,, and. Exhaustion conjecture…

Let $M$ be a Seifert $3$-manifold with base orbifold $B$, and let $G=\pi_1(M)$. Suppose that the orbifold Euler characteristic satisfies $\chi(B)<0$. Negative-characteristic decida…

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…

Let $Y$ be a small Seifert fibered space with complementary legs and $e_0\geq 0$, equipped with a contact structure $\xi$ that is not balanced. The relevant contact structures are…

The $\mathcal{QHB}$ conjecture. If $Y$ admits a symplectic rational homology ball filling, 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 … be a Seifert fibered space with $\frac{p_i}{q_i}>1$ for every $i$ and with generalized Euler invariant $\varepsilon(Y)>0$. Euler-invariant conjecture. If $Y$ smoothly embeds…

Let $Y$ be a Seifert fibered space in the setting where smooth embedding in $S^4$ requires $Y$ to be partitionable, with complementary classes defined by the partitions of the mult…

Integral Seifert surgery conjecture. Seifert surgeries on non-trivial knots are integral, except for some cases.

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…

Let $K'$ be a hyperbolic knot in $S^3$, let $\gamma$ be a surgery slope, and let $M=K'(\gamma)$. A genus $2$ Heegaard splitting of $M$ is as in Theorem. Non-integral surgery conjec…

Let $M$ be an oriented irreducible 3-manifold with infinite fundamental group $\pi_1(M)$ and nontrivial center. Center conjecture. Then $M$ is a Seifert fibre space. This is an imm…

Non-integral Seifert surgery conjecture. A Seifert fibered space can never be obtained by non-integral Dehn surgery on a hyperbolic knot in $S^3$.