60 problems

Let $f:S\to\mathbb P^2$ be a generic cover, with branch curve $B_f$. Two generic covers $f:S_i\to\mathbb P^2$, $i=1,2$, are equivalent when they have the same branch curve and are…

Let $L$ be a fibered link in $S^3$. A simple branched cover is a map $\pi\colon S^3\to S^3$ branched over a link $L'$, and $\alpha$ and $\beta$ are unknots that are braid axes of…

Let $D\subset\mathbb{CP}^2$ be an algebraic singular plane curve with cusps and nodes, and let $\theta:\pi_1(\mathbb{CP}^2-D)\to S_N$ be a symmetric-group-valued monodromy morphism…

Let $K$ be a prime alternating knot, and let $D$ be an alternating diagram of $K$ without nugatory crossings, so that $D$ is a minimal diagram. Let $X^{(2)}_K$ denote the double co…

Let $K$ be a prime satellite knot in $S^3$, and for each positive integer $n$ let $Sigma_n(K)$ denote the $n$-fold cyclic branched cover of $S^3$ branched over $K$. A Gordon–Lidman…

Greene's conjecture. If a pair of links have homeomorphic branched double-covers, then either both are alternating or both are non-alternating.

Let $G$ be a primitive genus $g$ group, meaning a primitive subgroup of $S_n$ admitting elements $x_1,\ldots,x_r\in G$ such that … … and … where $\operatorname{ind} x_i$ is the min…

Let $K$ be a knot in $bS^3$, let $bbSigmad_2K$ be its 2-fold branched covering, and let $X(bbSigmad_2K)$ be the $bSL_2(bC)$ character variety of $bpi_1(bbSigmad_2K)$. Write…

Let $K$ be a knot in $S^3$, and let $(S^3,K)^d$ denote the $d$-fold branched cover of $S^3$ along $K$, where $d\geq 1$. Smith conjecture. The fundamental group … is nontrivial unle…

Quasi-adjunction eigenspace conjecture. Then $h^{n,0}_{\chi}(\bar U_G)=0$ unless the lift of $\chi$ belongs to a contributing global face of quasi-adjunction $\mathcal F$; in the l…

Let $(X,\omega)$ be a simply connected symplectic $4$-manifold. For sufficiently large $k$, let $G_k^0$ be the relevant subgroup associated with the branch curve of the degree-…

Let $(X,\omega)$ be a simply connected symplectic $4$-manifold. For the degree-$N_k$ symplectic branched covering associated with sufficiently large $k$, let $G_k^0$ be the relevan…

Let $f:X\to\mathbb{CP}^2$ be a sufficiently ample simply connected branched cover, with branch curve $D\subset\mathbb{CP}^2$. Let $[\omega]$ and $K_X$ denote the symplectic and can…

Let $F$ be a smooth $2$-knot, let $n(F)$ denote its odd Khovanov invariant, and let $\Sigma(F)$ denote the branched double-cover associated to $F$. Miller–Wong's conjecture. For ev…

Let $F$ be a smooth $2$-knot, and let $n(F)$ be the integer-valued odd Khovanov invariant associated to a dotted broken surface diagram of $F$. Let $\Sigma(F)$ denote the branched…

Let $L$ be a link and let $\mathbf{p}$ be a set of basepoints containing exactly one basepoint in each component of $L$. Write $\mathsf{\Sigma}_2(S^3,L)$ for the double branched co…

Cremona-normal form conjecture. Up to a Cremona transformation of the plane, one can assume that $D_{11}$ is irreducible, has an ordinary point of multiplicity $d-2$ at $p$, and ha…

The discussion concerns three local balance conditions: the condition for Speiser graphs in Theorem, the condition in section S, and the condition in section S. Equivalence conject…

Let $K$ be a knot or link and let $\Sigma(K)$ denote its double branched cover, equipped with a real structure $\iota$ and a compatible real $\mathrm{spin^c}$ structure…

Várilly-Alvarado and Vakil's limiting conjecture. For all $B \subseteq \mathbb{R}^{d-1}$, the limit $\lim_{g \to \infty}\pi^{\textnormal{geo}}(B,g)$ exists and depends only on the…

The Tschirnhausen realization conjecture. The tuple $(e_1,\dots,e_{d-1})$ is achieved as the scrollar invariants of a smooth irreducible primitive cover $C \to \mathbb{P}^1$ of deg…

Higher-degree transitivity conjecture. With appropriate modifications for any Hurwitz class, an analogous result to Theorem should hold for post-critically finite branched covers o…

Let $K\subset S^4$ be a smooth 2-knot. Removing a small neighborhood of a point of $K$ gives a smooth link cobordism from the empty link to the unknot $U$, and hence a map … For de…

Let $(X, \omega)$ be a translation surface, and let an orbit closure mean the $\operatorname{SL}_2(\mathbb{R})$-orbit closure of a translation surface in the moduli space. A branch…

Greene–Owens cubiquity conjecture. $\Sigma(L)$ bounds a rational homology $4$-ball if and only if $\Lambda(L)$ is cubiquitous.