Intrinsically Knotted and 4-Linked Directed Graphs

Thomas Fleming; Joel Foisy

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arXiv · 1702.06233

Intrinsically Knotted and 4-Linked Directed Graphs

Thomas Fleming·Joel Foisy

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Abstract

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 $G$ and its corresponding symmetric digraph $\overline{DG}$. Finally, we note that the maximum number of edges for a graph that is not intrinsically linked is $O(n)$ in the undirected case, but $O(n^2)$ for directed graphs.

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Thomas Fleming, Joel Foisy. 2017-02-21. Intrinsically Knotted and 4-Linked Directed Graphs. https://arxiv.org/abs/1702.06233

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