33 problems

A Moishezon 4-manifold is the smooth 4-manifold constructed from a semi-algebraic cuspidal curve in $\mathbb P^2$ and an epimorphism to a symmetric group, with a smooth map to…

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 $F$ 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…

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 $L$ 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…

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…

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 $K\subset S^4$ 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…

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

Let $K$ be a transverse knot in $S^3$, let $\Sigma_2(K)$ be its double branched cover, and let $\widetilde{\xi}$ be the induced contact structure. Let … and … be the corresponding…

Toroidal link conjecture. All cyclic branched covers $cSigma_{\psi}(L)$ of $L$ are $NLS$, $LO$, and $CTF$.

The $ADE$ Link Conjecture. If $L$ is not an $ADE$ link, then all cyclic branched covers $cSigma_{\psi}(L)$ of $L$ are $NLS$, $LO$, and $CTF$.

Branched-cover conjecture. For every $k>0$ there is $d$ such that, whenever $Q$ is a finite cubical poset of dimension $k+1$ and every $k$-face is contained in at least $d$ facets,…

For a link $L$ in $S^3$, let $M_L\to S^3$ be the $2$-fold branched cover branched over $L$. The quantity $\ell_g(M_L)$ denotes the minimal entropy associated with $M_L$. Universal…

Let $K$ be a 2-bridge knot, and let $Y=\Sigma(S^3,K)$ be its double branched cover. Let $\widetilde{\sigma}$ be a lift of the strong negative amphichiral involution when one exists…

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 $L$ be a prime, fibred, strongly quasipositive link. Write $k(L)$ for its BKL-exponent, defined as the maximum integer $k$ such that $L$ is the closure of $\delta_n^kP$ for som…

Let $L$ be a link in $S^3$, and for each integer $n\geq 2$ let $\Sigma_n(L)$ denote the canonical $n$-fold cyclic cover branched over $L$. A link is simply laced arborescent if it…

Universal abelian cover conjecture. If the universal abelian cover of $X(K)(n\mu+q\lambda)$ is not excellent, then $K$ is fibred and

The branched-cover L-space conjecture. If some $\Sigma_n(L)$ is an L-space, then $L$ is simply laced arborescent. If $L=K$ is a knot, then $K$ is a $(2,k)$, $(3,4)$, or $(3,5)$ tor…

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…