## Tokyo Journal of Mathematics

### Counting Links and Knots in Complete Graphs

#### Abstract

We investigate the minimal number of links and knots in embeddings of complete partite graphs in $S^3$. We provide exact values or bounds on the minimal number of links for all complete partite graphs with all but 4 vertices in one partition, or with 9 vertices in total. In particular, we find that the minimal number of links in an embedding of $K_{4,4,1}$ is 74. We also provide exact values or bounds on the minimal number of knots for all complete partite graphs with 8 vertices.

#### Article information

Source
Tokyo J. Math., Volume 36, Number 2 (2013), 429-458.

Dates
First available in Project Euclid: 31 January 2014

https://projecteuclid.org/euclid.tjm/1391177980

Digital Object Identifier
doi:10.3836/tjm/1391177980

Mathematical Reviews number (MathSciNet)
MR3161567

Zentralblatt MATH identifier
1285.05036

#### Citation

ABRAMS, Loren; MELLOR, Blake; TROTT, Lowell. Counting Links and Knots in Complete Graphs. Tokyo J. Math. 36 (2013), no. 2, 429--458. doi:10.3836/tjm/1391177980. https://projecteuclid.org/euclid.tjm/1391177980

#### References

• L. Abrams, B. Mellor and L. Trott, Gordian (Java computer program). Available at http://myweb.lmu.edu/bmellor/research/Gordian
• P. Blain, G. Bowlin, J. Hendricks, J. LaCombe and J. Foisy, Knotted Hamiltonian cycles in spatial embeddings of complete graphs, New York J. Math. 13 (2007), 11–16.
• P. Blain, G. Bowlin, T. Fleming, J. Foisy, J. Hendricks and J. LaCombe, Some results on intrinsically knotted graphs, J. Knot Theory Ramif. 16 (2007), 749–760.
• J. C. Cha and C. Livingston, KnotInfo: Table of Knot Invariants, http://www.indiana.edu/$\sim$knotinfo, May 11, 2010.
• J. Conway and C. Gordon, Knots and links in spatial graphs, J. of Graph Theory 7 (1983), 445–453.
• E. Flapan, Intrinsic knotting and linking of complete graphs, Algebraic and Geometric Topology 2 (2002), 371–380.
• E. Flapan, J. Foisy, R. Naimi and J. Pommersheim, Intrinsically $n$-linked graphs, J. Knot Theory Ramif. 10 (2001), 1143–1154.
• E. Flapan, B. Mellor and R. Naimi, Intrinsic linking is arbitrarily complex, Fund. Math. 201 (2008), 131–148.
• T. Fleming and B. Mellor, Counting links in complete graphs, Osaka J. Math. 46 (2009), 1–29.
• J. Foisy, Intrinsically knotted graphs, J. Graph Theory 39 (2002), 178–187.
• J. Foisy, A newly recognized intrinsically knotted graph, J. Graph Theory 43 (2003), 199–209.
• J. Foisy, More intrinsically knotted graphs, J. Graph Theory 54 (2007), 115–124.
• J. Foisy and L. Ludwig, When graph theory meets knot theory, in Communicating mathematics, pp. 67–85, Contemp. Math. 479, Amer. Math. Soc., Providence, RI, 2009.
• R. K. Guy, Latest results on crossing numbers, in Recent Trends in Graph Theory, Springer, New York, 1971, pp. 143–156.
• Y. Hirano, Improved lower bound for the number of knotted hamiltonian cycles in spatial embeddings of complete graphs, J. Knot Theory Ramif. 19 (2010), 705–708.
• B. Johnson and W. Johnson, On the size of links in $K_{n,n}$, $K_{n,n,1}$ and $K_n$, J. Knot Theory Ramif. 11 (2002), 145–152.
• T. Kohara and S. Suzuki, Some remarks on knots and links in spatial graphs, Knots 90 (Osaka, 1990), de Gruyter, Berlin, 1992, pp. 435–445.
• R. Motwani, A. Raghunathan and H. Saran, Constructive results from graph minors: linkless embeddings, $29$th Annual Symposium on Foundations of Computer Science, IEEE, 1988, pp. 398–409.
• R. Nikkuni and K. Taniyama, $\triangle Y$-exchanges and the Conway-Gordon theorems, preprint, April 2011, arXiv:1104.0828.
• N. Robertson, P. Seymour and R.Thomas, Sachs' linkless embedding conjecture, J. Combin. Theory Ser. B 64 (1995), 185–227.