Intrinsically knotted and 4-linked directed graphs

We consider intrinsic linking and knotting in the context of directed graphs. We construct an example of a directed graph that contains a consistently oriented knotted cycle in every embedding. We also construct examples of intrinsically 3-linked and 4-linked directed graphs. We introduce two operations, consistent edge contraction and H-cyclic subcontraction, as special cases of minors for digraphs, and show that the property of having a linkless embedding is closed under these operations. We analyze the relationship between the number of distinct knots and links in an undirected graph [Formula: see text] and its corresponding symmetric digraph [Formula: see text]. Finally, we note that the maximum number of edges for a graph that is not intrinsically linked is [Formula: see text] in the undirected case, but [Formula: see text] for directed graphs.

GRAPHS WITH A KNOT OR 3-COMPONENT LINK IN EVERY SPATIAL EMBEDDING

We describe a family of graphs that contain a knot or a non-split 3-component link in every spatial embedding. We exhibit a graph in this family that has a knotless embedding, and a 3-component linkless embedding.