90 problems

Cabling Conjecture. If $K$ is a knot in $S^3$ which has a reducible surgery, then $K$ is a cabled knot and the reducing slope is given by the cabling annulus.

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…

A cosmetic surgery is a pair of Dehn fillings on a knot or 3-manifold exterior that produce homeomorphic filled manifolds; it is truly cosmetic when the homeomorphism preserves the…

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…

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 $M$ be a cusped manifold, and let $(W_n,L_n,\rho_n)$ be a sequence of compact hyperbolic Dehn fillings of $M$ converging geometrically to $M$, where $L_n$ consists of the simpl…

For an integer $p$, define … Let $M_p$ be the closed three-manifold obtained from the three-sphere by $p$-surgery along the figure-eight knot. Stationary-phase conjecture for figur…

Let $K$ be a nontrivial L-space knot in $S^3$, and let $g(K)$ be its Seifert genus. The conjugate-slope property at $n$ is the property defined in the paper for $n\in\mathbb N$. Co…

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 $M$ be a cusped hyperbolic manifold with at least one cusp, and let $\mathbf{s}$ and $\mathbf{s'}$ be sets of slopes on the cusps. Futer–Purcell–Schleimer's conjecture. If the…

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…

Trivial-character skein-module conjecture. If $X(M)$ is finite, then

Boileau's nontrivial surgery conjecture. A non-trivial surgery on $K$ never gives a manifold homeomorphic to $Y$ in an orientation-preserving way.

Let $K$ be a knot in an L-space. The knot $K$ is persistently foliar if, except for one meridional slope, every boundary slope of its complement is strongly realized by a co-orient…

Unbounded-slope conjecture. For any natural number $N$ there exist pairwise-different integers $n_1,\ldots,n_N$ and non-isotopic knots $K$ and $K'$ such that, for $i=1,\ldots,N$,…

Cosmetic surgery conjecture. If $p/q\neq p'/q'$, then $S^3_{p/q}(K)$ and $S^3_{p'/q'}(K)$ are not orientation-preserving diffeomorphic.

Let $K$ be a knot in $S^3$ with Seifert genus $g>0$. Let $\mathcal{S}_K\subset\mathbb{Q}$ be the set of rational slopes $s$ for which there exists a taut $\mathcal{C}^0$-foliation…

Let $\Lambda$ be a non-trivial Legendrian knot that is the Legendrian rainbow closure of a positive braid. For $k\in {\mathbb N}_+$, let $(S^{3}_{1/k}(\Lambda),\xi_{1/k}(\Lambda))$…

Let $K$ be a knot in $S^3$ that is not a $(2,r)$ torus knot, and let $N=S^3-K$. Let $(s,s')$ be a chirally cosmetic pair of slopes on $\partial N$. Ichihara–Ito–Saito's conjecture.…

A veering triangulation $\Delta$ on an oriented $3$-manifold $N$ determines a transitive pseudo-Anosov flow $\phi^t(\Delta,M)$ on a closed oriented $3$-manifold $M$, and a transiti…

Let $\phi^t$ be a pseudo-Anosov flow on a closed oriented $3$-manifold, let $c$ be a positive/negative horizontal surgery curve, and let $n$ be a positive/negative integer, respect…

Let $K$ be a hyperbolic knot in $S^3$ that admits non-trivial exceptional surgeries. A NIT boundary slope is a boundary slope that is non-integral or toroidal. For $b e{}$ an integ…

Let $K$ be a hyperbolic knot in $S^3$. A boundary slope is a slope represented by the boundary curves of an essential embedded surface in the exterior of $K$, and an exceptional su…

Let $K$ be a knot in $S^3$. Two surgeries on $K$ with distinct slopes $r,r'\in\mathbb{Q}\cup\{\infty\}$ are chirally cosmetic surgeries when $S^3_r(K)\cong -S^3_{r'}(K)$, where the…