## Involve: A Journal of Mathematics

• Involve
• Volume 1, Number 2 (2008), 145-158.

### On graphs for which every planar immersion lifts to a knotted spatial embedding

#### Abstract

We call a graph $G$ intrinsically linkable if there is a way to assign over/under information to any planar immersion of $G$ such that the associated spatial embedding contains a pair of nonsplittably linked cycles. We define intrinsically knottable graphs analogously. We show there exist intrinsically linkable graphs that are not intrinsically linked. (Recall a graph is intrinsically linked if it contains a pair of nonsplittably linked cycles in every spatial embedding.) We also show there are intrinsically knottable graphs that are not intrinsically knotted. In addition, we demonstrate that the property of being intrinsically linkable (knottable) is not preserved by vertex expansion.

#### Article information

Source
Involve, Volume 1, Number 2 (2008), 145-158.

Dates
Accepted: 1 December 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.involve/1513799086

Digital Object Identifier
doi:10.2140/involve.2008.1.145

Mathematical Reviews number (MathSciNet)
MR2429655

Zentralblatt MATH identifier
1154.57003

#### Citation

DeCelles, Amy; Foisy, Joel; Versace, Chad; Wilson, Alice. On graphs for which every planar immersion lifts to a knotted spatial embedding. Involve 1 (2008), no. 2, 145--158. doi:10.2140/involve.2008.1.145. https://projecteuclid.org/euclid.involve/1513799086

#### References

• C. C. Adams, The knot book. An elementary introduction to the mathematical theory of knots, American Mathematical Society, Providence, RI, 2004. Revised reprint of the 1994 original.
• J. H. Conway and C. M. Gordon, “Knots and links in spatial graphs”, J. Graph Theory 7:4 (1983), 445–453.
• M. R. Fellows and M. A. Langston, “Nonconstructive tools for proving polynomial-time decidability”, J. Assoc. Comput. Mach. 35:3 (1988), 727–739.
• J. Foisy, “Intrinsically knotted graphs”, J. Graph Theory 39:3 (2002), 178–187. http://www.ams.org/mathscinet-getitem?mr=2003a:05051MR 2003a:05051
• L. H. Kauffman, Formal knot theory, Mathematical Notes 30, Princeton University Press, Princeton, NJ, 1983.
• R. Motwani, A. Raghunathan, and H. Saran, “Constructive results from graph minors: Linkless embeddings”, pp. 398–409 in 29th Annual Symposium on Foundations of Computer Science, IEEE, 1988.
• J. Nešetřil and R. Thomas, “A note on spatial representation of graphs”, Comment. Math. Univ. Carolin. 26:4 (1985), 655–659.
• N. Robertson and P. D. Seymour, “Graph minors. XX. Wagner's conjecture”, J. Combin. Theory Ser. B 92:2 (2004), 325–357.
• N. Robertson, P. Seymour, and R. Thomas, “Sachs' linkless embedding conjecture”, J. Combin. Theory Ser. B 64:2 (1995), 185–227.
• H. Sachs, “On a spatial analogue of Kuratowski's theorem on planar graphs–-an open problem”, pp. 230–241 in Graph theory (Łagów, 1981), edited by M. Borowiecki et al., Lecture Notes in Math. 1018, Springer, Berlin, 1983.
• I. Sugiura and S. Suzuki, “On a class of trivializable graphs”, Sci. Math. 3:2 (2000), 193–200.
• N. Tamura, “On an extension of trivializable graphs”, J. Knot Theory Ramifications 13:2 (2004), 211–218.
• K. Taniyama, “Knotted projections of planar graphs”, Proc. Amer. Math. Soc. 123:11 (1995), 3575–3579.
• K. Taniyama and A. Yasuhara, “Realization of knots and links in a spatial graph”, Topology Appl. 112:1 (2001), 87–109.