57 problems

Let $L$ be a knot or link type, let ${\operatorname{Cr}}(L)$ denote its crossing number, and let ${\operatorname{Rib}}([L])$ denote the infimal folded ribbonlength of the knot or l…

Let $G$ be a graph, let $\textrm{cr}(G)$ be its crossing number, let $\chi(G)$ be its chromatic number, and let $K_r$ be the complete graph on $r$ vertices. Albertson's conjecture.…

Ichihara's conjecture. For all knots $K$,

Crossing-number conjecture. The crossing number of a satellite knot cannot be less than the crossing number of its companion knot. This conjecture would make the bound in the prece…

Let $p,q$ be positive integers with $\gcd(p,q)=1$, and let $K$ be the $(p,q)$-cable knot about a knot $C$ in $\mathcal{F}$. Cable crossing-number conjecture. The crossing number sh…

Positive-diagram crossing-number conjecture.

Crossing-number equality conjecture.

Parallel crossing number conjecture.

Asymptotic crossing number conjecture.

Parallel overcrossing number conjecture. The parallel overcrossing number of $K$ equals $\operatorname{Cr}(K)$.

Harary's crossing-number conjecture. The displayed upper bound is tight for every $r$; equivalently,

Qazaqzeh–Mansour's conjecture. In this case, the crossing number satisfies

Let $K$ be a knot type, let $c[K]$ denote its crossing number, and let $\operatorname{stick}[K]$ denote its stick number. Universal linear stick-number conjecture. Every knot with…

Let $c[K]$ denote the crossing number of a knot type $K$, and let $\operatorname{stick}[K]$ denote its stick number. Let $\beta_*$ be the asymptotic constant such that, for every…

Strong one-crossing 4-coloring conjecture. If $G$ has no separating triangles and every vertex not incident with the crossed edges has degree at least five, then $G$ is 4-colorable…

Minimum-degree conjecture. Every 5-critical graph in ${\cal C}$ contains a vertex of degree four.

For a rectilinear drawing of the complete graph $K_n$, let $e_k(K_n)$ denote the number of edges crossed exactly $k$ times, and let … denote the maximum of $e_k(K_n)$ over all rect…

Let $G$ be a complete graph or a complete bipartite graph, let $S_t$ be the orientable surface of genus $t$, and define its algebraic genus by … Here $\delta_t(G)$ is the Euler exc…

Positive stable crossing growth conjecture.

Stable crossing number conjecture. The stable crossing number exists and

Satellite crossing number conjecture. The following inequalities are conjectured:

Stable crossing growth conjecture. The crossing number grows like

Satellite knot crossing-number conjecture. One has

Satellite crossing-number conjecture. For every satellite knot $K$ with companion knot $k$,

Let $L_1$ and $L_2$ be links, and let $c(L)$ denote the crossing number of a link $L$. Their connected sum is a link obtained by joining $L_1$ and $L_2$ along arcs. Crossing-number…