Skip to main content
Springer Nature Link
Log in

A partial order in the knot table II

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

A partial order on the set of the prime knots can be defined by the existence of a surjective homomorphism between knot groups. In the previous paper, we determined the partial order in the knot table. In this paper, we prove that 31 and 41 are minimal elements. Further, we study which surjection a pair of a periodic knot and its quotient knot induces, and which surjection a degree one map can induce.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Whitten, W.: Knot complements and groups. Topology, 26, 41–44 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  2. Gordon, C., Luecke, J.: Knots are determined by their complements. J. Amer. Math. Soc., 2(2), 371–415 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  3. Kawauchi, A. (ed): A Survey of Knot Theory, Birkhaüser Verlag, Basel, 1996

    MATH  Google Scholar 

  4. Kitano, T., Suzuki, M.: A partial order on the knot table. Experimental Math., 14, 385–390 (2005)

    MATH  MathSciNet  Google Scholar 

  5. Rolfsen, D.: Knots and links, AMS Chelsea Publishing

  6. Neuwirth, L.: Knot Groups, Annals of Mathematics Studies, No. 56 Princeton University Press, Princeton, N.J., 1965

    MATH  Google Scholar 

  7. Stallings, J.: On fibering certain 3-manifolds, 1962 Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) 95–100 Prentice-Hall, Englewood Cliffs, N.J.

    Google Scholar 

  8. Silver, D., Whitten, W.: Knot group epimorphisms. J. Knot Theory Ramifications, 15, 153–166 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hempel, J.: 3-manifolds, Reprint of the 1976 original. AMS Chelsea Publishing, Providence, RI, 2004

    MATH  Google Scholar 

  10. Papakyriakopoulos, C. D.: On Dehn’s lemma and the asphericity of knots. Ann. of Math. (2), 66, 1–26 (1957)

    Article  MathSciNet  Google Scholar 

  11. Crowell, R., Fox, R.: Introduction to knot theory, GTM 57, Springer

  12. Kitano, T., Suzuki, M., Wada, M.: Twisted Alexander polynomial and surjectivity of a group homomorphism. Algebr. Geom. Topol., 5, 1315–1324 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kodama, K.: http://www.math.kobe-u.ac.jp/HOME/kodama/knot.html

  14. Kodama, K., Sakuma, M.: Symmetry groups of prime knots up to 10 crossings. Knot, 90, 323–340 (1990)

    Google Scholar 

  15. Murasugi, K.: On periodic knots. Comment. Math. Helv., 46, 162–174 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  16. Wada, M.: Twisted Alexander polynomial for finitely presentable groups. Topology, 33, 241–256 (1994)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Information Systems Science, Faculty of Engineering, Soka University, 1-236 Tangi-cho, Hachioji-city, Tokyo, 192-8577, Japan

    Teruaki Kitano

  2. Department of Mathematics, Akita University, 1-1 Tegata-Gakuenmachi, Akita, 010-8502, Japan

    Masaaki Suzuki

Authors
  1. Teruaki Kitano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Masaaki Suzuki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Teruaki Kitano.

Additional information

The authors are supported in part by Grand-in-Aid for Scientific Research (No. 17540064 and No. 18840008)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kitano, T., Suzuki, M. A partial order in the knot table II. Acta. Math. Sin.-English Ser. 24, 1801–1816 (2008). https://doi.org/10.1007/s10114-008-6269-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-008-6269-2

Keywords

MR(2000) Subject Classification

Search

Navigation