24 problems

Positive-diagram crossing-number conjecture.

Parallel crossing number conjecture.

Tait's conjecture. A reduced alternating diagram of $L$ has minimal crossing number, and if $L$ is prime then no non-alternating diagram has minimal crossing number.

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…

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…

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…

Positive stable crossing growth conjecture.

Eventual linearity conjecture. For sufficiently large $n$,

Stable crossing number conjecture. The stable crossing number exists and

Let $L_1\subset\Sigma_1\times I$ and $L_2\subset\Sigma_2\times I$ be links in thickened surfaces, and let $D_1,D_2$ be minimal-crossing representatives of $L_1,L_2$, respectively.…

Potential crossing pair conjecture. The graph $G$ has crossing number at least $2$ if and only if it does not have a potential crossing pair.

Universal deviation inequality. For every face $F\in\mathcal{F}(D)$,

A graph is $k$-crossing-critical if its crossing number is at least $k$ while deleting any edge reduces the crossing number below $k$. Let $g:\mathbb N\to\mathbb R^+$ be a function…

Let $\operatorname{cr}(X)$ denote the crossing number of a knot $X$. A prime knot $P$ is $\lambda$-regular if … whenever $P$ is a factor of a knot $K$. Two-thirds regularity conjec…

Let $K$ be a composite knot with factors $P_1,\ldots,P_r$, and let $\operatorname{cr}(X)$ denote the crossing number of a knot $X$. Composite-knot factor crossing-number conjecture…

Let $K$ be a satellite knot and let $P$ be its companion. Let $\operatorname{cr}(X)$ denote the crossing number of a knot $X$. Satellite-knot crossing-number conjecture. The crossi…

Let $\tau_1$ and $\tau_2$ be knots, let \tau_1#\tau_2 denote their connected sum, and let $\operatorname{cr}(K)$ denote the crossing number of a knot $K$. Crossing-number additiv…

Let $G$ be a graph with a given rotation system, and let $k\geq 1$ be an integer. A drawing of $G$ respecting the prescribed rotation system is one in which the clockwise order of…

Let $G$ be a graph, and let $e$ and $f$ be Kuratowski edges of $G$, meaning that each belongs to a subgraph homeomorphic to $K_{3,3}$ or $K_5$. A graph obtained by multiplying edge…

Let $G$ be a graph. An edge of $G$ is a Kuratowski edge if it belongs to a subgraph homeomorphic to $K_{3,3}$ or $K_5$. Kuratowski-edge conjecture. If every edge of $G$ is a Kurato…

A graph is $k$-crossing-critical if its crossing number is at least $k$, but every proper subgraph has crossing number smaller than $k$. High-degree vertex conjecture. For every po…

A graph is $k$-crossing-critical if its crossing number is at least $k$, but every proper subgraph has crossing number smaller than $k$. Richter and Salazar's conjecture. For every…

Let $T_n$ denote the degree-$n$ Chebyshev polynomial of the first kind. For an odd integer $i\ge 3$, the difference-two crossing-number conjecture. The parametrization … parametriz…

Let $T_n$ denote the degree-$n$ Chebyshev polynomial of the first kind. For an odd integer $i\ge 3$, the consecutive-degree crossing-number conjecture. The parametrization … parame…