Advertisement

Journal of Mathematical Chemistry

, Volume 52, Issue 1, pp 23–41 | Cite as

The Homfly polynomial of double crossover links

  • Xiao-Sheng Cheng
  • Yujuan Lei
  • Weiling Yang
Original Paper

Abstract

Motivated by double crossover DNA polyhedra (He et al. in Nature 452:198, 2008; Lin et al. in Biochemistry 48:1663, 2009; Zhang et al. in J Am Chem Soc 131:1413, 2009; Zhang et al. in Proc Natl Acad Sci USA 105:10665, 2008; He et al. in Angew Chem Int Ed 49:748, )2010, in this paper, we construct a new type of link, called the double crossover link, formed by utilizing the “\(n\)-point star” to cover each vertex of a connected graph \(G\). The double crossover link can be used to characterize the topological properties of double crossover DNA polyhedra. We show that the Homfly polynomial of the double crossover link can be obtained from the chain polynomial of the truncated graph of \(G\) with two distinct labels. As an application, by using computer algebra (Maple) techniques, the Homfly polynomial of a double crossover tetrahedral link is obtained. Our result may be used to characterize and analyze the topological structure of DNA polyhedra.

Keywords

DNA polyhedron Polyhedral links Homfly polynomial  Chain polynomial 

Notes

Acknowledgments

X.-S. Cheng was supported by Grants from the National Natural Science Foundation of China (No. 11101174), Natural Science Foundation of Guangdong Province of China (No. S2011040003984) and Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (No. LYM11120). Y. Lei and W. Yang was supported by Grants from the National Natural Science Foundation of China (No. 11271307), Natural Science Foundation of Fujian Province of China (No. 2012J01019).

References

  1. 1.
    Y. He, T. Ye, M. Su, C. Zhang, A.E. Ribbe, W. Jiang, C. Mao, Nature 452, 198 (2008)CrossRefGoogle Scholar
  2. 2.
    C. Lin, Y. Liu, H. Yan, Biochemistry 48, 1663 (2009)CrossRefGoogle Scholar
  3. 3.
    C. Zhang, S.H. Ko, M. Su, Y. Leng, A.E. Ribbe, W. Jiang, C. Mao, J. Am. Chem. Soc. 131, 1413 (2009)CrossRefGoogle Scholar
  4. 4.
    C. Zhang, M. Su, Y. He, X. Zhao, P. Fang, A.E. Ribbe, W. Jiang, C. Mao, Proc. Natl. Acad. Sci. USA 105, 10665 (2008)CrossRefGoogle Scholar
  5. 5.
    Y. He, M. Su, P. Fang, C. Zhang, A.E. Ribbe, W. Jiang, C. Mao, Angew. Chem. Int. Ed. 49, 748 (2010)CrossRefGoogle Scholar
  6. 6.
    J.P. Sauvage, C.D. Buckecker, Molecular Catenanes, Rotaxanes and Knots (Wiley, New York, 1999)CrossRefGoogle Scholar
  7. 7.
    C.A. Schalley, Angew. Chem. Int. Ed. 43, 4399 (2004)CrossRefGoogle Scholar
  8. 8.
    J.S. Siegel, Science 304, 1256 (2004)CrossRefGoogle Scholar
  9. 9.
    O. Lukin, F. Vögtle, Angew. Chem. Int. Ed. 44, 1456 (2005)CrossRefGoogle Scholar
  10. 10.
    B.C. Dietrich, B.X. Colasson, J.P. Sauvage, Molecular Knots: In Templates in Chemistry Ii (Springer, Berlin, 2005)Google Scholar
  11. 11.
    C.D. Pentecost, K.S. Chichak, A.J. Peters, G.W.V. Cave, S.J. Cantrill et al., Angew. Chem. Int. Ed. 46, 218 (2007)CrossRefGoogle Scholar
  12. 12.
    P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millett et al., Bull. Am. Math. Soc. (N.S.) 12, 239 (1985)CrossRefGoogle Scholar
  13. 13.
    J.H. Przytycki, P. Traczyk, Kobe J. Math. 4, 115 (1987)Google Scholar
  14. 14.
    X. Jin, F. Zhang, MATCH Commun. Math. Comput. Chem. 63, 657 (2010)Google Scholar
  15. 15.
    S.Y. Liu, H. Zhang, W.Y. Qiu, MATCH Commun. Math. Comput. Chem. 67, 65 (2012)Google Scholar
  16. 16.
    S.Y. Liu, X.S. Cheng, H. Zhang, W.Y. Qiu, J. Math. Chem. 48, 439 (2010)CrossRefGoogle Scholar
  17. 17.
    X. Jin, F. Zhang, Proc. Am. Math. Soc. 140, 1459 (2012)CrossRefGoogle Scholar
  18. 18.
    G. Hu, W.Y. Qiu, A. Ceulemans, PLoS One 6, e26308 (2011)CrossRefGoogle Scholar
  19. 19.
    X.S. Cheng, X. Jin, PLoS One 7, e48968 (2012)CrossRefGoogle Scholar
  20. 20.
    P.G. Cromwell, Knots and Links (Cambridge University Press, Cambridge, 2004)CrossRefGoogle Scholar
  21. 21.
    L. Traldi, Proc. Am. Math. Soc. 106, 279 (1989)CrossRefGoogle Scholar
  22. 22.
    L.H. Kauffman, Discrete Appl. Math. 25, 105 (1989)CrossRefGoogle Scholar
  23. 23.
    R.C. Read, E.G. Whitehead, Discrete Math. 204, 337 (1999)CrossRefGoogle Scholar
  24. 24.
    R.C. Read, E.G. Whitehead Jr, Discrete Math 243, 267 (2002)CrossRefGoogle Scholar
  25. 25.
    X. Jin, F. Zhang, Advances in Appl. Math. 34, 47 (2005)CrossRefGoogle Scholar
  26. 26.
    X. Jin, F. Zhang, J. Stat. Mech. P07011. (2011)Google Scholar
  27. 27.
    W.T. Tutte, Can. J. Math. 6, 80 (1954)CrossRefGoogle Scholar
  28. 28.
    F. Jaeger, Proc. Am. Math. Soc. 103, 647 (1988)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsHuizhou UniversityHuizhouPeople’s Republic of China
  2. 2.School of Mathematical SciencesXiamen UniversityXiamenPeople’s Republic of China

Personalised recommendations