167 problems

Let $K$ be a nontrivial knot, and let $K(r)$ denote the result of $r$-Dehn surgery on $K$ for $r\in\mathbb{Q}$. Property P conjecture. The manifold $K(r)$ is never simply connected…

Let $K$ be a knot in $S^3$ and let $r,s e r$ be slopes in $mathbb{Q}cup{infty}$. Write $S^3r(K)$ for the manifold obtained by Dehn surgery on $K$ with slope $r$; two surgeries are…

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

A knot $K$ is chiral if it is not equivalent to its mirror image, and $T(2,n)$ denotes the $(2,n)$ torus knot. Chirally cosmetic surgery conjecture. If $K$ is a chiral knot that is…

Let $K$ be a knot in a lens space $L$ and let surgery on $K$ produce a non-prime $3$-manifold $Y$. A knot $K$ is hyperbolic if its exterior admits a complete finite-volume hyperbol…

Let $K$ be a knot in $S^3$. A surgery on $K$ is reducible if the resulting 3-manifold is reducible, and $K$ is a cabled knot if it is obtained as a cable of another knot. The cabli…

Let $K\subset S^3$ be a knot, and let $c$ be a crossing of $K$. A crossing is nugatory if its crossing circle bounds a disk in the complement of $K$; a generalized crossing change…

Let $N$ be a finite-volume hyperbolic 3-manifold with one or more cusps. Let $\mathbf{s}$ and $\mathbf{s}'$ be tuples of slopes on the cusps of $N$. Hyperbolic cosmetic surgery con…

Let $K$ be a non-trivial knot in $S^3$. A pair of Dehn surgeries on $K$ with inequivalent slopes is chirally cosmetic if the resulting $3$-manifolds are orientation-reversingly hom…

Chinburg–Reid–Stover's conjecture. In the notation above,

Classical L-space knot conjecture. The knot $kappa$ is persistently foliar if and only if it has no non-trivial L-space or reducible surgeries.

Let $K$ be a knot in a closed connected orientable $3$--manifold $Y$, with exterior $M=Y\setminus\operatorname{int}N(K)$ irreducible and not homeomorphic to the solid torus. A slop…

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.

Cosmetic Surgery Conjecture. Under these hypotheses, $K$ admits no purely cosmetic surgeries.

Let $N$ be a cusped hyperbolic 3-manifold, and consider fillings of $N$ along distinct slopes that produce hyperbolic manifolds. A filling pair is cosmetic if the resulting manifol…

Maximal exceptional-slope distance conjecture. The maximal intersection number between exceptional slopes is $8$, and this maximum is realized by the figure-eight knot complement.

The $L(4,1)$ surgery conjecture. The lens space $L(4,1)$ cannot be obtained from a non-trivial knot in $S^3$ by Dehn surgery.

Fiberedness conjecture for hyperbolic knots yielding lens spaces. If a hyperbolic knot admits Dehn surgery yielding a lens space, then the knot is fibered.

Let $K$ be a knot in a homotopy sphere, let $M$ be its exterior, and let $M_\gamma$ denote the Dehn filling obtained by gluing a solid torus to $\partial M$ so that the simple clos…

Let $K\subset S^3$ be a hyperbolic knot, and suppose that a surgery on $K$ yields a lens space of order $r$. Bleiler--Litherland's conjecture. The order satisfies … The paper prove…

Let $K\subset S^3$ be a hyperbolic knot of genus $g$, and suppose that a surgery on $K$ yields a lens space of order $r$. Goda--Teragaito's conjecture. The order satisfies … The co…

Let $K$ be a hyperbolic knot in $S^3$. A surgery slope on $K$ is toroidal if the corresponding Dehn surgery contains an incompressible torus. Eudave-Muñoz's conjecture. The knot…

A knot manifold is an irreducible, orientable, compact 3-manifold whose boundary is a single torus. It is hyperbolic if its interior admits a complete hyperbolic metric of finite v…

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.