22 problems

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…

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 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…

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…

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…

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…

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…

Suppose that $X^4$ is a closed $4$--manifold admitting a Lefschetz fibration whose regular fibers are tori. Let $T\subset X$ be a regular fiber of the fibration, and suppose that…

Baker's conjecture. If $K$ is hyperbolic, then

Let $m$ and $n$ be positive integers, and consider the Seifert-fibered dihedral manifolds … A knot surgery is a Dehn surgery on a knot in the $3$-sphere, and the family is consider…

A finite manifold $Y$ is a rational homology sphere with finite fundamental group, and $d(Y,\mathfrak{t})$ denotes its Heegaard Floer correction term for a $b\mathfrak{t}d$ spin…

Let $L(p,q)$ be a lens space, let $K\subset L(p,q)$ be a knot, let $g(K)$ denote its genus, and let $P^3$ denote the Poincaré sphere. An integer surgery is Dehn surgery with an int…

Let $S^1\times D^2$ be the solid torus and let $S^1\times S^2$ be the product of the circle with the 2-sphere. An integer lens space surgery is integer surgery on a knot in…

Knot-and-link generation conjecture. All distinct exotic smoothness structures of the compact 4-manifold $M=E(2)$ can be generated by knots and links.

Realization conjecture. Then $L(p,q)$ is realized by integer surgery on a Berge knot.