No Access

Intrinsic linking and knotting in tournaments

Thomas Fleming

http://orcid.org/0000-0003-2947-4587

666 5th Avenue, 8th Floor, New York, NY 10103, USA

Department of Mathematics, SUNY Potsdam, Potsdam, NY 13676, USA

Search for more papers by this author

and

Joel Foisy

666 5th Avenue, 8th Floor, New York, NY 10103, USA

Department of Mathematics, SUNY Potsdam, Potsdam, NY 13676, USA

Search for more papers by this author

https://doi.org/10.1142/S0218216519500767Cited by:1 (Source: Crossref)

Abstract

A directed graph $G$ is intrinsically linked if every embedding of that graph contains a nonsplit link $L$ , where each component of $L$ is a consistently oriented cycle in $G$ . A tournament is a directed graph where each pair of vertices is connected by exactly one directed edge.

We consider intrinsic linking and knotting in tournaments, and study the minimum number of vertices required for a tournament to have various intrinsic linking or knotting properties. We produce the following bounds: intrinsically linked ( $n = 8$ ), intrinsically knotted ( $9 \leq n \leq 12$ ), intrinsically 3-linked ( $10 \leq n \leq 23$ ), intrinsically 4-linked ( $12 \leq n \leq 66$ ), intrinsically 5-linked ( $15 \leq n \leq 154$ ), intrinsically $m$ -linked ( $3 m \leq n \leq 8 {(2 m - 3)}^{2}$ ), intrinsically linked with knotted components ( $9 \leq n \leq 107$ ), and the disjoint linking property ( $12 \leq n \leq 14$ ).

We also introduce the consistency gap, which measures the difference in the order of a graph required for intrinsic $n$ -linking in tournaments versus undirected graphs. We conjecture the consistency gap to be nondecreasing in $n$ , and provide an upper bound at each $n$ .

Keywords:

AMSC: 57M15, 57M25, 05C10, 05C20

References

1. L. Abrams, B. Mellor and L. Trott, Counting links and knots in complete graphs, Tokyo J. Math. 36(2) (2013) 429–458. Crossref, Web of Science, Google Scholar
2. S. Chan, A. Dochtermann, J. Foisy, J. Hespen, E. Kunz, T. Lalonde, Q. Loney, K. Sharrow and N. Thomas, Graphs with disjoint links in every spatial embedding, J. Knot Theory Ramifications 13(6) (2004) 737–748. Link, Web of Science, Google Scholar
3. J. H. Conway and C. McA. Gordon, Knots and links in spatial graphs, J. Graph Theory 7 (1983) 446–453. Crossref, Web of Science, Google Scholar
4. E. Flapan, J. Foisy, R. Naimi and J. Pommersheim, Intrinsically n-linked graphs, J. Knot Theory Ramification 10(8) (2001) 1143–1154. Link, Web of Science, Google Scholar
5. E. Flapan, B. Mellor and R. Naimi, Intrinsic linking and knotting are arbitrarily complex, Fund. Math. 201 (2008) 131–148. Crossref, Web of Science, Google Scholar
6. E. Flapan, R. Naimi and J. Pommershiem, Intrinsically triple linked complete graphs, Topol. Appl. 115 (2001) 239–246. Crossref, Web of Science, Google Scholar
7. T. Fleming, Intrinsically linked graphs with knotted components, J. Knot Theory Ramifications 21(7) (2012) 1250065-1–1250065-10. Link, Web of Science, Google Scholar
8. T. Fleming and J. Foisy, Intrinsically knotted and 4-linked directed graphs, J. Knot Theory Ramification 27(6) (2018) 1850037-1–1850037-18. Link, Web of Science, Google Scholar
9. T. Fleming and B. Mellor, Counting links in complete graphs, Osaka J. Math. 46 (2009) 173–201. Web of Science, Google Scholar
10. J. Foisy, Intrinsically knotted graphs, J. Graph Theory 39(3) (2002) 178–187. Crossref, Web of Science, Google Scholar
11. J. Foisy, H. Howards and N. Rich, Intrinsic linking in directed graphs, Osaka J. Math. 52 (2015) 817–831. Web of Science, Google Scholar
12. R. Hanaki, R. Nikkuni, K. Taniyama and A. Yamazaki, On intrinsically knotted or completely 3-linked graphs, Pac. J. Math. 252 (2011) 407–425. Crossref, Web of Science, Google Scholar
13. T. Mattman, R. Naimi and B. Pagano, Intrinsic linking and knotting are arbitrarily complex in directed graphs, arXiv:1901.01212. Google Scholar
14. H. Sachs, On Spatial Representations of Finite Graphs, eds. A. Hajnal, L. Lovasz and V. T. Sós, Colloquium Mathematical Society János Bolyai, Vol. 37 (North-Holland, Amsterdam, 1984), pp. 649–662. Google Scholar
15. K. Taniyama and A. Yasuhara, Realization of knots and links in a spatial graph, Topol. Appl. 112 (2001) 87–109. Crossref, Web of Science, Google Scholar