29 problems

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…

Classification conjecture. The invariant is a complete invariant of the set of equivalence classes modulo surgery in $\ker\rho$. Equivalently, the theorem asserting that there are…

Distinctness conjecture. All the tight contact structures defined by Figure and satisfying the assumptions are distinct up to isotopy.

Suppose $n_x>0$, $n_y<0$, and $n_z>0$, and let $M_{n_x,n_y,n_z}$ carry the Anosov flows $\phi_{xy}$ and $\phi_{yz}$ from the two surgery descriptions. A contact-structure swapping…

A positive/negative horizontal surgery curve for a pseudo-Anosov flow is a steady curve of the corresponding sign, and horizontal Goodman surgery is defined by cutting along a tran…

Let $\Phi_1$ and $\Phi_2$ be suspension pseudo-Anosov flows. A Dehn-Goodman-Fried surgery is a surgery performed on a periodic orbit of a pseudo-Anosov flow to produce another pseu…

Let $L$ be a link with a well-defined GM series, and let $L'$ be obtained by partial $-1/r$ surgery on an unknot component, or by $\infty$-surgery in the corresponding case. Partia…

The regularization conjecture. The invariant is given by

Let $\widehat{Z}(q)$ be the invariant associated with a 3-manifold obtained by $+1$-surgery, and consider its singularities on the unit circle $|q|=1$. The singularity-at-$q=1$ con…

Casals–Etnyre–Kegel conjecture. If the contact $(+1)$-surgeries and contact $(-1)$-surgeries on two Legendrian knots are contactomorphic, then the knots are Legendrian isotopic.

Let $L$ be a Legendrian knot in $S^3$ with its standard tight contact structure. Two contact surgeries on a Legendrian knot are cosmetic if the resulting contact manifolds are cont…

A horizontal surgery operation on veering triangulations modifies a veering triangulation along a horizontal surgery curve, while horizontal Goodman surgery performs the correspond…

Generalized cosmetic crossing conjecture. No knot admits a generalized cosmetic crossing.

Let $Y$ be a closed, oriented $3$-manifold and let $K$ be a framed knot in $Y$. Let $\mathcal H$ be a nice bordered Heegaard diagram for the infinity-framed complement…

Let $K\subset S^3$ be a knot, let $S^3_{-n}(K)$ be obtained by $(-n)$-surgery, and let $W$ be the natural cobordism from $S^3$ to $S^3_{-n}(K)$. Let $t_i:H_2(W_{-n})\to\mathbb{Z}$…

Let $M=\mathcal{W}\cup_\mathcal{T}\mathcal{V}$, where $\mathcal{T}=\partial \mathcal{W}=\partial \mathcal{V}=\mathcal{W}\cap\mathcal{V}$ is a genus-$g$ surface. Let…

Let $K\subset S^3$ be a knot, let $p$ be fixed, and let $S^3_{p/q}(K)$ denote $p/q$ surgery, with $q$ tending to infinity through integers relatively prime to $p$. For each degree…

Let $L$ be a $g$-component link embedded as a cut system on a genus-$g$ Heegaard surface in $\#^j(S^1 \times S^2)$, and suppose that surface-framed surgery on $L$ yields…

Let $Y$ be an integer homology sphere containing a knot $K$ which admits a surgery to $S^1 \times S^2$. Kirby's generalized Property R conjecture. Then $K$ is unique up to ambient…

Seifert surgery conjecture. Seifert surgeries on non-trivial knots are integral, except in some cases, and the resultant Seifert fibered spaces have three singular fibers.

Let $L$ be a null-homologous Legendrian knot with Thurston–Bennequin invariant $tb(L)\leq -2$, and let $n$ be a positive integer. Overtwistedness conjecture. If $n<|tb(L)|$, then c…

Let $K$ be a hyperbolic knot in $S^3$, and let $p>0$ be an integer such that the $p$-surgery on $K$ is a spherical space form of $\mathbf{T}$-, $\mathbf{O}$-, or $\mathbf{I}$-type.…

Doubly primitive knot classification conjecture. If $K$ is hyperbolic, then

Mauricio's conjecture. Surgery on $K$ never produces

Gonzales-Acuña–Short's cabling conjecture. If a knot in $S^3$ admits a surgery to a non-prime $3$-manifold, then the knot is either a torus knot or a cabled knot.