33 problems

Let $L$ be a knot or link in $S^3$. Its bridge index is the minimum number of local maxima of a diagram of $L$ with respect to a height function, and its meridional rank is the min…

Let $\Gamma_1$ and $\Gamma_2$ be Brunnian $\theta$-curves, meaning spatial $\theta$-curves in $S^3$ whose proper subgraphs are unknotted. Suppose their complements…

Let $G$ be a $2$-connected graph, and let $\Gamma$ be a non-trivial spatial graph of $G$. Define the representativity of $\Gamma$ by … where $\mathcal{F}$ is the set of all closed…

Let an oriented link be a boundary link if its components bound pairwise disjoint Seifert surfaces in $S^3$. Let the trivial link be the link whose components are disjoint unknotte…

Let oriented links be related by self $\Delta$-equivalence when one can be transformed into the other by self delta moves and ambient isotopies, where each self delta move has all…

Let $G$ be a graph, and let $mnl(G)$ denote the minimum number of links among spatial embeddings of $G$. A book embedding is a spatial embedding in which the graph is embedded with…

Let $K_{m,n}$ be a complete bipartite graph with independent vertex sets $\{a_1,\dots,a_m\}$ and $\{b_1,\dots,b_n\}$. Place $a_i$ at $(i,0)$ on the $x$-axis and $b_j$ at $(0,j)$ on…

The degree-zero/degree-one conjecture.

Injectivity conjecture. If two classical spatial graphs are virtually equivalent, then they are classically equivalent.

Let $G$ be a graph. Call $G$ intrinsically knotted if every classical spatial embedding of $G$ contains a non-trivial knot, and call it intrinsically virtually knotted of degree 1…

A classical spatial graph is a spatial graph represented by a classical graph diagram, and two such graphs are virtually equivalent or classically equivalent according to the virtu…

Let $G,H\subset S^3$ be spatial graphs, let $G\circ H$ denote their loop sum, and let $tp(G)$ denote the three-page complexity of a spatial graph. Three-page complexity additivity…

Let $S(p,q)$ be the non-oriented $2$-bridge link with parameters $p,q\geq 1$, and let $tp(G)$ denote the minimal value of $ar(G)-2$ over all general three-page embeddings of a spat…

Let $\Gamma$ be a spatial graph, and let $S$ be the boundary of a tubular neighborhood of $\Gamma$. The spatial-graph DGA is the DGA constructed from a diagram of $\Gamma$ by the e…

Hamiltonian spine-selection conjecture. There exists a choice of Hamiltonian spine and corresponding Hamiltonian leveled embedding of $\mathcal{G}$ for which the Hamiltonian levele…

Leveled-spine conjecture. A cycle can be found that is the spine of a leveled embedding of $\mathcal{G}$ for which the algorithm ends successfully.

Spine-selection conjecture. There is a choice of spine for which the algorithm ends successfully.

Cellular-embedding conjecture for connected leveled embeddings. All connected leveled embeddings can be cellular embedded with the presented algorithm.

Main Conjecture. A spatial graph $G$ has a finite $N$-quandle if and only if there is a spherical orbifold with underlying space $\mathbb{S}^3$ whose singular locus is the edge-lab…

Mellor–Smith classification conjecture. Every labeled link with finite $N$-quandle is the singular locus of a spherical orbifold with underlying space $\mathbb{S}^3$ appearing in D…

Minor-closedness conjecture. Having every possible conflict graph balanced for every maximally planar subgraph $M$ is closed under minors.

Flat-embedding conjecture. The graph $G$ has a flat embedding precisely when every possible conflict graph is balanced for every maximally planar subgraph $M$. Moreover, if one max…

Intrinsic-linking conjecture. A graph $G$ is intrinsically linked if and only if every maximal planar subgraph of $G$ has every possible conflict graph unbalanced.

Let $\Gamma$ be a finite connected graph, and let its freeness index be the largest integer measuring how many edges may be deleted while the complement of a suitable embedding of…

A finite connected graph $\Gamma$ has freeness index at least $2$ if it admits an embedding in $S^3$ such that, after deleting any one edge, the complement of the embedded graph is…