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Intrinsic linking and knotting are arbitrarily complex in directed graphs

Mattman Thomas W., Naimi Ramin, Pagano Benjamin

Bulletin of the Polish Academy of Sciences. Mathematics

2021

Vol. 69, no. 1

1--9

Fleming and Foisy (2018) recently proved the existence of a digraph whose every embedding contains a 4-component link, and left open the possibility that a directed graph with an intrinsic n-component link might exist. We show that, indeed, this is the case. In fact, much as Flapan, Mellor, and Naimi (2008) show for graphs, knotting and linking are arbitrarily complex in directed graphs. Specifically, we prove the analog for digraphs of the main theorem of their paper: for any n and α, every embedding of a sufficiently large complete digraph in R3 contains an oriented link with components Q1,…,Qn such that, for every i≠j, |lk(Qi,Qj)|≥α and |a2(Qi)|≥α, where a2(Qi) denotes the second coefficient of the Conway polynomial of Qi.