Skip to main content
Log in

Abstract

It is well known that a shaded link diagram corresponds to a signed plane multi-graph. In graph theory, line graph is an old and important concept originally introduced by H. Whitney in 1932. In this paper we define the line graph link to be a link which has a diagram whose corresponding signed plane graph is a signed line graph. The main purpose of the paper is to illustrate the structure of planar line graphs, which permits us to deal with its signed Tutte polynomial and the Jones polynomials of line graph links.

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

Access this article

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. Adams, C.C. The knot book: an elementary introduction to the mathematical theory of knots. AMS Providence, Rhode Island, 2004

    MATH  Google Scholar 

  2. Beineke, L.W., Wilson, R.J. Selected Topics in Graph Theory, Vol. 1. Academic Press Inc, London, 1978

    MATH  Google Scholar 

  3. Chang, S.C., Shrock, R. Zeros of Jones polnomials for families of knots and links. Physica A, 301: 196–218 (2001)

    Article  MathSciNet  Google Scholar 

  4. Cheng, X.S., Liu, S.Y., Zhang, H.P., Qiu, W.Y. Fabrication of a family of pyramidal links and their genus. MATCH Commun. Math. Comput. Chem., 63: 623–636 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Godsil, C., Royle, G. Algebraic graph theory. Springer-Verlag, New York, 2001

    Book  Google Scholar 

  6. Harris, A.B. Possible Néel orderings of the Kagomé antiferromagnet. Phys. Rev. B, 45, 2899–2919 (1992)

    Article  Google Scholar 

  7. Hu, G., Qiu, W.Y. Extended Goldberg polyhedra. MATCH Commun. Math. Comput. Chem., 59: 585–594 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Jin, X., Zhang, F. The Kauffman brackets for equivalence classes of links. Advances in Appl. Math., 34: 47–64 (2005)

    Article  MathSciNet  Google Scholar 

  9. Jin, X., Zhang, F. Zeros of the Jones polynomial for multiple crossing-twisted links. J. Stat. Physics, 140(6): 1054–1064 (2010)

    Article  MathSciNet  Google Scholar 

  10. Jin, X., Ge, J., Cheng, X.S., Lin, Y. The number of circles of a maximum state of a plane graph with applications. Acta Mathematicae Applicatae Sinica, 37(2): 409–420(2021)

    Article  MathSciNet  Google Scholar 

  11. Jin, X., Zhang, F. On computing Kauffman bracket polynomial of Montesinos links. Journal of Knot Theory and its Ramifications, 19(8): 1001–1023 (2010)

    Article  MathSciNet  Google Scholar 

  12. Kaufman, L.H. Knots and Physics. World Scientific Publishing Co, Singapore, 1993

    Google Scholar 

  13. Kauffman, L.H. A Tutte polynomial for signed graphs. Discrete Appl. Math., 25: 105–127 (1989)

    Article  MathSciNet  Google Scholar 

  14. Krausz, J. Démonstration nouvelle d’un théorème de Whitney sur les réseaux. Mat. Fiz. Lapok, 50: 75–85 (1943)

    MathSciNet  MATH  Google Scholar 

  15. Lander, G.C., Evilevitch, A., Jeembaeva, M., Potter, C.S., Carragher, B., Johnson, J.E. Bacteriophage Lambda Stabilization by Auxiliary Protein gpD: Timing, Location, and Mechanism of Attachment Determined by Cryo-EM. Structure, 16: 1399–1406 (2008)

    Article  Google Scholar 

  16. Liu, S., Zhang, H. Genera of the links derived from 2-connected plane graphs. J. Knot Theory Ramifications, 21: 1250129–1250143 (2012)

    Article  MathSciNet  Google Scholar 

  17. Lovasz, L., Plummer, M.D. Matching Theory, Elsevier Science, Amsterdam, 1986

    MATH  Google Scholar 

  18. Qiu, W.Y., Zhai, X.D. Molecular design of Goldberg polyhedral links. J. Mol. Struc. (Theochem) 756: 163–166 (2005)

    Article  Google Scholar 

  19. Qiu, W.Y., Zhai, X.D., Qiu, Y.Y. Architecture of Platonic and Archimedean polyhedral links. Science in China (Ser. B) Chemistry, 51(1): 13–18 (2008)

    Article  Google Scholar 

  20. Roussopoulos, N.D. A max {m, n} algorithm for determining the graph H from its line graph G. Information Processing Letters, 2(4): 108–112 (1973)

    Article  MathSciNet  Google Scholar 

  21. Whitney, H. Congruent graphs and the Connectivity of Graphs. Amer. J. Math., 54: 150–168 (1932)

    Article  MathSciNet  Google Scholar 

  22. Wikoff, W.R., Liljas, L., Duda, R.L., Tsuruta, H., Hendrix, R.W., Johnson, J.E. Topologically Linked Protein Rings in the Bacteriophage HK97 Capsid. Science, 289: 2129–2133(2000)

    Article  Google Scholar 

  23. Wikoff, W.R., Liljas, L., Duda, R.L., Tsuruta, H., Hendrix, R.W., Johnson, J.E. Topologically Linked Protein Rings in the Bacteriophage HK97 Capsid. Science, 289: 2129–2133(2000)

    Article  Google Scholar 

  24. Wu, F.Y., Wang, J. Zeros of the Jones polynomial. Physica A, 296: 483–494 (2001)

    Article  MathSciNet  Google Scholar 

  25. Yang, W., Zhang, F. Links and cubic 3-polytopes. Mathematics of Computation, 77(263): 1841–1857 (2008)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Wei-ling Yang.

Additional information

This paper is supported by the Fundamental Research Funds for the Central Universities (Grant No. 20720190071).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, Wl., Jin, Xa. & Zhang, Fj. Line Graph Links. Acta Math. Appl. Sin. Engl. Ser. 37, 706–716 (2021). https://doi.org/10.1007/s10255-021-1041-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-021-1041-9

Keywords

2000 MR Subject Classification

Navigation