14 problems

Let $G$ be a $\theta$-graph in $S^3$. A belt $B$ is an oriented annulus intersecting the edges of $G$ in intervals, and $G(B)$ denotes the spatial graph obtained by buckling $G$ al…

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…

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

Decomposition criterion. The graphs $\Gamma$ and $\Gamma'$ are equivalent if and only if they admit decompositions

A polyhedral molecule has an underlying graph that is abstractly planar, 3-connected, and simple; a molecule is toroidal when its graph embedding on the standard torus does not emb…

Let $G$ be a planar graph embedded nontrivially on the torus. A nontrivial knot is a knotted embedded circle, and a nonsplit link is a nonsplit link with at least two components. C…

Minimal knotted-cycle conjecture. The canonical book representation $\widetilde{K}_n$ contains the fewest total number of knotted cycles possible in any embedding of $K_n$.

Let $K$ be an alternating knot, and let $r(K)$ denote its representativity. Alternating knot representativity conjecture. It holds that … The source gives no further discussion of…