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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.
Wydawca
Rocznik
Tom
Strony
1--9
Opis fizyczny
Bibliogr. 4 poz., rys.
Twórcy
autor
- Department of Mathematics and Statistics California State University, Chico Chico, CA 95929-0525, U.S.A
autor
- Department of Mathematics Occidental College Los Angeles, CA 90041, U.S.A.
autor
- Department of Mathematics Occidental College Los Angeles, CA 90041, U.S.A.
Bibliografia
- [1] J. Conway and C. Gordon, Knots and links in spatial graphs, J. Graph Theory 7 (1983), 445–453.
- [2] E. Flapan, B. Mellor, and R. Naimi, Intrinsic linking and knotting are arbitrarily complex, Fund. Math. 201 (2008), 131–148.
- [3] T. Fleming and J. Foisy, Intrinsically knotted and 4-linked directed graphs, J. Knot Theory Ramif. 27 (2018), art. 1850037, 18 pp.
- [4] H. Sachs, On spatial representations of finite graphs, in: Finite and Infinite Sets, Vol. II (Eger, 1981), Colloq. Math. Soc. János Bolyai 37, North-Holland, Amsterdam, 1984, 649–662.
Typ dokumentu
Bibliografia
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