Intrinsic linking and knotting are arbitrarily complex in directed graphs

• Autorzy: Mattman Thomas W. , Naimi Ramin , Pagano Benjamin

Identyfikatory

DOI

10.4064/ba210704-8-7

• Języki publikacji: EN

Abstrakty

EN

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(Q_i,Q_j)|≥α and |a₂(Q_i)|≥α, where a₂(Q_i) denotes the second coefficient of the Conway polynomial of Q_i.

Słowa kluczowe

EN

intrinsically knotted graph; intrinsically linked graph; directed graph; spatial graph

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2021
• Tom: Vol. 69, no. 1
• Strony: 1--9
• Opis fizyczny: Bibliogr. 4 poz., rys.

Twórcy

autor

Mattman Thomas W.

• Department of Mathematics and Statistics California State University, Chico Chico, CA 95929-0525, U.S.A

• https://orcid.org/0000-0002-4900-6783

autor

Naimi Ramin

• Department of Mathematics Occidental College Los Angeles, CA 90041, U.S.A.

• https://orcid.org/0000-0002-8083-7833

autor

Pagano Benjamin

• Department of Mathematics Occidental College Los Angeles, CA 90041, U.S.A.

• https://orcid.org/0000-0001-8252-5377

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.