58 problems

A smoothly embedded projective plane in $S^4$ is knotted if it is not the standard unknotted embedding, and it is reducible if it admits an unknotted summand that is not a $2$-sphe…

Let $S$ be a closed nonorientable surface smoothly embedded in $\mathbb{S}^4$, and let $\chi$ be its Euler characteristic. Whitney's conjecture. The Euler number of $S$ is bounded…

Let $S$ be a closed, oriented surface smoothly embedded in $\mathbb{C}\mathbb{P}^2$. Write $d_a(S)$ for its algebraic degree and $g(S)$ for its genus. Topologically locally-flat Th…

Forest reconfiguration conjecture. A forest can always be reconfigured on any non-orientable surface.

For a smoothly embedded projective plane $F$ in $S^4$, its peripheral subgroup is the image of … A projective plane is knotted when it is not the standard unknotted embedding. Kata…

Let $\Sigma$ be a fixed surface. Let $G$ be a graph embeddable in $\Sigma$, and let $X,Y\subseteq V(G)$. An $X$-$Y$ path is a path with at least one end in $X$ and at least one end…

Let $S$ be a surface, and let $G$ be an excluded minor for $S$. The order of $G$ is bounded by a polynomial function of the Euler genus of $S$, and the paper establishes a bound of…

Toroidal shallow-minor conjecture. There is no constant $k\in\mathbb N$ with the following property: every toroidal graph is a congestion-$k$ depth-$k$ minor of a planar graph.

Surface transduction-order conjecture. Let $\Sigma$ and $\Gamma$ be surfaces such that $\Sigma\not\preccurlyeq\Gamma$. Then

Let $\Sigma$ be a surface and let $g(\Sigma)$ denote its genus. Fix an integer $k$ for which there is a value $\rho_0=\rho_0(g(\Sigma))$ such that every $\rho_0$-locally planar gra…

Let $G$ be a graph embedded in a surface $\Sigma$, and let $g(\Sigma)$ denote the genus of $\Sigma$. A graph is $\rho$-locally planar if the relevant local planarity condition hold…

Bounded-genus universal-obstruction conjecture. One has

Let $X$ be a $4$-manifold with finitely presented fundamental group, and consider the main theorem of the paper, which constructs smoothly distinct surfaces that remain distinct af…

Let $g$ be a nonnegative integer, and let $G$ be a graph of Euler genus at most $g$. A graph parameter is $(a,b)$-self-dual if its value on the dual graph is bounded in terms of it…

An $(n,t)$-quadrangulation is a quadrangular embedding of a simple graph on $n$ vertices and $\binom{n}{2}-t$ edges. An embedding is face-simple if its dual is simple, meaning that…

Let $A\in H_2(\mathbb CP^2,\mathbb Z)$, and let $\Sigma$ be an embedded surface representing $A$. The genus of $\Sigma$ is compared among all embedded surfaces representing the sam…

Cabaniss and Jackson's conjecture. For all $n\geq 11$,

Jackson and Ringel's conjecture. For all $g\geq 1$, the bound in this inequality is tight:

Fix a simply-connected smooth $4$-manifold $N$ with boundary $S^3$, a knot $K$, and a genus $g\textgreater=0$. For each $(N,K,g)$-realisable nondegenerate Hermitian form $(H,bb)$ o…

Genus-square-root conjecture. If $G$ is a graph of genus $g$, then

Let $g$ be a genus function and let $\mathcal A^g$ be one of the labelled graph classes considered in the paper. Logarithmic-threshold growth-constant conjecture. If $c>0$ is a con…

Let $g$ be a genus function, and let $\mathcal A^g$ denote one of the graph classes considered in the paper, such as $\mathcal E^g$, $\mathcal O\mathcal E^g$,…

Let $\mathcal A^h$ be one of the graph classes considered in the paper, indexed by graphs on $[n]$ embeddable in surfaces of Euler genus at most $h$. Genus-increment growth-ratio c…

Let $S$ be a surface, and let $d4cE^S_n$ denote the class of graphs on $[n]$ embeddable in $S$. Let $d4cP$ be the class of planar graphs, with planar growth constant…

Let $\jump(g,r)$ be the maximum choice–chromatic gap among graphs embeddable on the orientable surface of genus $g$ with chromatic number $r$, and let $H(g)$ be the maximum chromat…