Skip to main content
SpringerLink
Log in

Spatial graphs with local knots

  • Published:
Revista Matemática Complutense Aims and scope Submit manuscript

Abstract

It is shown that for any locally knotted edge of a 3-connected graph in S 3, there is a ball that contains all of the local knots of that edge which is unique up to an isotopy setwise fixing the graph. This result is applied to the study of topological symmetry groups of graphs embedded in S 3.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chambers, D., Flapan, E., O’Brien, J.: Topological symmetry groups of K 4r+3. Discrete Contin. Dyn. Syst. 4, 1401–1411 (2011)

    MathSciNet  MATH  Google Scholar 

  2. Flapan, E.: Rigidity of graph symmetries in the 3-sphere. J. Knot Theory Ramif. 4, 373–388 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Flapan, E., Mellor, B., Naimi, R.: Complete graphs whose topological symmetry groups are polyhedral. Algebr. Geom. Topol. (to appear)

  4. Flapan, E., Naimi, R., Pommersheim, J., Tamvakis, H.: Topological symmetry groups of embedded graphs in the 3-sphere. Comment. Math. Helv. 80, 317–354 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Flapan, E., Naimi, R., Tamvakis, H.: Topological symmetry groups of complete graphs in the 3-sphere. J. Lond. Math. Soc. 73, 237–251 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hashizume, Y.: On the uniqueness of the decomposition of a link. Osaka Math. J. 10, 283–300 (1958)

    MathSciNet  Google Scholar 

  7. Jaco, W., Shalen, P.: Seifert Fibred Spaces in 3-Manifolds. Memoirs of American Mathematical Society, vol. 220. Amer. Math. Soc., Providence (1979)

    Google Scholar 

  8. Johannson, K.: Homotopy Equivalences of 3-Manifolds with Boundaries. Lecture Notes in Mathematics, vol. 761. Springer, New York (1979)

    MATH  Google Scholar 

  9. Schubert, H.: Die eindeutige Zerlegbarkeit eines knots in prime knoten. S. B. Heidelberger Akad. Wiss. Math. Natur. Kl., pp. 57–104 (1949)

  10. Simon, J.: Topological chirality of certain molecules. Topology 25, 229–235 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  11. Smith, P.A.: Transformations of finite period II. Ann. Math. 40, 690–711 (1939)

    Article  Google Scholar 

  12. Suzuki, S.: A prime decomposition theorem for a graph in the 3-sphere. In: Topology and Computer Science, pp. 259–276. Kinokuniya, Tokyo (1987)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Pomona College, Claremont, CA, 91711, USA

    Erica Flapan

  2. Department of Mathematics, Loyola Marymount University, Los Angeles, CA, 90045, USA

    Blake Mellor

  3. Department of Mathematics, Occidental College, Los Angeles, CA, 90041, USA

    Ramin Naimi

Authors
  1. Erica Flapan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Blake Mellor
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ramin Naimi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Erica Flapan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Flapan, E., Mellor, B. & Naimi, R. Spatial graphs with local knots. Rev Mat Complut 25, 493–510 (2012). https://doi.org/10.1007/s13163-011-0072-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13163-011-0072-9

Keywords

Mathematics Subject Classification (2000)

Search

Navigation