34 problems

Let $K\subset S^3$ be a lens space knot, meaning that some positive integer surgery on $K$ produces a lens space. Berge's construction gives a family of such knots. Berge conjectur…

Let $K$ and $K'$ be knots in $S^3$. Their zero surgeries are the $3$-manifolds $S^3_0(K)$ and $S^3_0(K')$, respectively, and the knots are concordant when they cobound a smooth ann…

In the Artin spun case, let knots be called essentially different when they are not isotopic, are not mirrors of one another, and are not isotopic under mirroring of connect-summan…

Thin surgery discrepancy question. Does there exist a strongly invertible L-space knot $(K,\Phi)$ with a thin surgery such that $\varkappa(K,\Phi)$ is not thin?

Let $E(2)$ be the elliptic K3 surface, and let $X_{K_1}$ and $X_{K_2}$ denote the manifolds obtained from $E(2)$ by knot surgery using knots $K_1$ and $K_2$, respectively. Knot-sur…

For every $5\leq m\leq 8$, let $R_m$ be a manifold homeomorphic to $P_m$, constructed so that it has vanishing Seiberg–Witten invariants and contains a nullhomologous torus with th…

Let $K$ be a knot in $S^3$. Suppose there exists a rational number $r$ such that the $3$-manifold obtained by $r$-surgery on $K$ is homeomorphic to the Poincaré homology sphere…

Let $f_-$ be the map defined by the holomorphic triangles in the surgery construction, and let $\delta_1$ be the differential component described in the comparison with Ozsváth–Sza…

Let $M$ be a knot-surgery manifold homotopic to a $K3$ surface, obtained from a knot $K$. Knot-surgery fibredness conjecture. If $K$ is not a fibered knot, then $M$ cannot be sympl…

Fintushel–Stern injectivity conjecture. The association

Let a Stein trace be the Stein $4$-manifold obtained from a Legendrian knot by attaching a Stein $2$-handle. Contact annulus twists modify the knot and its contact data, while Lutt…

Let $K$ and $K'$ be knots in $S^3$, and let $S^3_0(K)$ denote the $3$-manifold obtained by zero surgery on $K$. A knot is H-slice in a $4$-manifold $X$ if it bounds a smoothly embe…

Let $K$ and $K'$ be knots in $S^3$, and let $S^3_0(K)$ denote the $3$-manifold obtained by zero surgery on $K$. A knot is slice if it bounds a smoothly embedded disk in $B^4$. Slic…

Thin-slope formal semi-group question. Is every sufficiently large slope of $K$ a thin slope if and only if the formal semi-group of $K$ is not a semi-group?

Instanton L-space surgery conjecture. Under these assumptions, $S^3_{p/q}(K)$ must be an instanton L-space, meaning

Let $K$ be a knot in $S^3$. A lensbordant surgery on $K$ is a positive integer surgery that is smoothly rational homology cobordant to a lens space, and let $u^+(K)$ denote the kn…

Characterizing slopes conjecture. There exists a constant $C=C(K)$ such that any slope $p/q$ satisfying $|q|\geq 2$ and $|p|+|q|\geq C$ is characterizing for $K$.

Suppose $\hat{\mu}=q\mu+p\lambda$ with $q>0$, and suppose $\hat{\lambda}=q_0\mu+p_0\lambda$ is chosen as in the setup. For $m\in\mathbb{Z}$ such that $\hat{\lambda}-m\hat{\mu}$ is…

The bypass-map conjecture. The displayed square commutes for every $s\in[-g(K),g(K)]$. This predicts that the bypass-map summand is identified with the corresponding basic-class su…

Let $K'$ be a knot in a homology sphere $\Sigma$, and let $K=C_{p,q}(K')$ be its $(p,q)$-cable. The $pq$-surgery on $K$ satisfies … When $\Sigma=S^3$, the cabling conjecture. This…

Let $K\subset S^3$ be a knot, and for a rational number $r$ let $S^3_r(K)$ denote the $r$-surgery on $K$. Stipsicz's high surgery conjecture. There is an integer $n_K$ such that fo…

Consider an initial spherical universe $M=S^3$ that undergoes knot surgery, producing the Poincaré dodecahedral space behind a spherical event horizon associated with a collapsed t…

Let $L(p,q)$ be a lens space with universally tight contact structure $\xi_{UT}$, and let $K\subset L(p,q)$ be a knot admitting a surgery to the 3-sphere. A Legendrian representati…

Let $p>1$ and $q>0$ be relatively prime integers. Let $P(p,q)$ be the prism manifold associated with these parameters, let $K$ be a knot in $S^3$, and let $\mathcal P^+$ denote the…

Let $P$ be a pattern in the solid torus, let $K$ be a knot, let $U$ denote the unknot, and write $P(K)$ and $P(U)$ for the corresponding satellite knots. An L-space knot is a knot…