Skip to main content
SpringerLink
Log in

The architecture and the Jones polynomial of polyhedral links

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

In this paper, we first recall some known architectures of polyhedral links (Qiu and Zhai in J Mol Struct (THEOCHEM) 756:163–166, 2005; Yang and Qiu in MATCH Commun Math Comput Chem 58:635–646, 2007; Qiu et al. in Sci China Ser B Chem 51:13–18, 2008; Hu et al. in J Math Chem 46:592–603, 2009; Cheng et al. in MATCH Commun Math Comput Chem 62:115–130, 2009; Cheng et al. in MATCH Commun Math Comput Chem 63:115–130, 2010; Liu et al. in J Math Chem 48:439–456 2010). Motivated by these architectures we introduce the notions of polyhedral links based on edge covering, vertex covering, and mixed edge and vertex covering, which include all polyhedral links in Qiu and Zhai (J Mol Struct (THEOCHEM) 756:163–166, 2005), Yang and Qiu (MATCH Commun Math Comput Chem 58:635–646, 2007), Qiu et al. (Sci China Ser B Chem 51:13–18, 2008), Hu et al. (J Math Chem 46:592–603, 2009), Cheng et al. (MATCH Commun Math Comput Chem 62:115–130, 2009), Cheng et al. (MATCH Commun Math Comput Chem 63:115–130, 2010), Liu et al. (J Math Chem 48:439–456, 2010) as special cases. The analysis of chirality of polyhedral links is very important in stereochemistry and the Jones polynomial is powerful in differentiating the chirality (Flapan in When topology meets chemistry. Cambridge Univ. Press, Cambridge, 2000). Then we give a detailed account of a result on the computation of the Jones polynomial of polyhedral links based on edge covering developed by Jin, Zhang, Dong and Tay (Electron. J. Comb. 17(1): R94, 2010) and, at the same time, by using this method we obtain some new computational results on polyhedral links of rational type and uniform polyhedral links with small edge covering units. These new computational results are helpful to judge the chirality of polyhedral links based on edge covering. Finally, we give some remarks and pose some problems for further study.

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. Chen J., Seeman N.C.: Synthesis from DNA of a molecule with the connectivity of a cube. Nature 350, 631–633 (1991)

    Article  CAS  Google Scholar 

  2. Goodman R.P., Berry R.M., Turberfield A.J.: The single-step synthesis of a DNA tetrahedron. Chem. Commun. 12, 1372–1373 (2004)

    Article  Google Scholar 

  3. Shih W.M., Quispe J.D., Joyce G.F.: A 1.7-kilobase single-stranded DNA that folds into a nanoscale octahedron. Nature 427, 618–621 (2004)

    Article  CAS  Google Scholar 

  4. Zhang Y., Seeman N.C.: The construction of a DNA truncated octahedron. J. Am. Chem. Soc. 116, 1661–1669 (1994)

    Article  CAS  Google Scholar 

  5. Erben C.M., Goodman R.P., Turberfield A.J.: A self-assembled DNA bipyramid. J. Am. Chem. Soc. 129, 6992–6993 (2007)

    Article  CAS  Google Scholar 

  6. Zimmermann J., Cebulla M.P.J., Mönninghoff S., Kiedrowski G.V.: Self-assembly of a DNA dodecahedron from 20 trisoligonucleotides, with C 3h linkers. Angew. Chem. Int. Ed. 47, 3626–3630 (2008)

    Article  CAS  Google Scholar 

  7. He Y., Ye T., Su M., Zhang C., Ribbe A.E., Jiang W., Mao C.D.: Hierarchical self-assembly of DNA into symmetric supramolecular polyhedra. Nature 452, 198–202 (2008)

    Article  CAS  Google Scholar 

  8. 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  CAS  Google Scholar 

  9. Helgstrand C., Wikoff W.R., Duda R.L., Hendrix R.W., Johnson J.E., Liljas L.: The refined structure of a protein catenane: the HK97 bacteriophage capsid at 3.44 Å resolution. J. Mol. Biol. 334, 885–899 (2003)

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  11. Yang Y.M., Qiu W.Y.: Molecular design and mathematical analysis of carbon nanotube links. MATCH Commun. Math. Comput. Chem. 58, 635–646 (2007)

    CAS  Google Scholar 

  12. Qiu W.Y., Zhai X.D., Qiu Y.Y.: Architecture of platonic and Archimedean polyhedral links. Sci. China Ser. B Chem. 51, 13–18 (2008)

    Article  CAS  Google Scholar 

  13. Hu G., Zhai X.D., Lu D., Qiu W.Y.: The architecture of platonic polyhedral links. J. Math. Chem. 46, 592–603 (2009)

    Article  CAS  Google Scholar 

  14. Cheng X.S., Qiu W.Y., Zhang H.P.: A novel molecular design of polyhedral links and their chiral analysis. MATCH Commun. Math. Comput. Chem. 62, 115–130 (2009)

    CAS  Google Scholar 

  15. 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, 115–130 (2010)

    Google Scholar 

  16. Liu S.Y., Cheng X.S., Zhang H.P., Qiu W.Y.: The architecture of polyhedral links and their HOMFLY polynomials. J. Math. Chem. 48, 439–456 (2010)

    Article  CAS  Google Scholar 

  17. Sumners D.W.: The knot theory of molecules. J. Math. Chem. 1, 1–14 (1987)

    Article  CAS  Google Scholar 

  18. Flapan E.: When Topology Meets Chemistry. Cambridge Univ. Press, Cambridge (2000)

    Google Scholar 

  19. Jin X., Zhang F.: The Homfly polynomial for even polyhedral links. MATCH Commun. Math. Comput. Chem. 63, 657–677 (2010)

    Google Scholar 

  20. Jin X., Zhang F.: The Jones polynomial for polyhedral links. MATCH Commun. Math. Comput. Chem. 65(2), 501–520 (2011)

    CAS  Google Scholar 

  21. Jin X., Zhang F., Dong F., Tay E.G.: Zeros of the Jones polynomial are dense in the complex plane. Electron. J. Comb. 17(1), R94 (2010)

    Google Scholar 

  22. Jaeger F., Vertigan D.L., Welsh D.J.A.: On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Camb. Phil. Soc. 108, 35–53 (1990)

    Article  Google Scholar 

  23. Landvoy R.A.: The Jones polynomial of pretzel knots and links. Topol. Appl. 83, 135–147 (1998)

    Article  Google Scholar 

  24. P.M.G. Manchón, On the Kauffman bracket of pretzel links. Marie Curie Fellowships Annals, Second Volume, (2002). http://www.mariecurie.org/annals/volume2/manchon.pdf

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

    Article  Google Scholar 

  26. Jin X., Zhang F.: The replacements of signed graphs and Kauffman brackets of link families. Adv. Appl. Math. 39, 155–172 (2007)

    Article  Google Scholar 

  27. Diao Y., Hetyei G., Hinson K.: Tutte polynomials of tensor products of signed graphs and their applications in knot theory. J. Knot Theory Ramif. 18(5), 561–590 (2009)

    Article  Google Scholar 

  28. Diao Y., Ernst C., Ziegler U.: Jones polynomial of knots formed by repeated tangle replacement operations. Topol. Appl. 156, 2226–2239 (2009)

    Article  Google Scholar 

  29. S.V. Jablan, Lj. Radović, R. Sazdanović, Tutte and Jones polynomial of link families, arXiv:1004.4302v1 [math.GT]

  30. Jablan S.V., Radović Lj., Sazdanović R.: Tutte and Jones polynomials of links, polyominoes and graphical recombination patterns. J. Math. Chem. 49, 79–94 (2011)

    Article  CAS  Google Scholar 

  31. Yang W., Zhang F.: Links and cubic 3-polytopes. Math. Comput. 77, 1841–1857 (2008)

    Article  Google Scholar 

  32. Wu F.Y.: Jones polynomial as a Potts model partition function. J. Knot Theory Ramif. 1(1), 47–57 (1992)

    Article  Google Scholar 

  33. Jones V.F.R.: Knot theory and statistical mechanics. Sci. Am 263, 98–103 (1990)

    Article  Google Scholar 

  34. Yang C.N., Lee T.D.: Statistical theory of equations of states and phase transitions. I. Theory of condensation. Phys. Rev. 87, 404–409 (1952)

    Article  CAS  Google Scholar 

  35. Lee T.D., Yang C.N.: Statistical theory of equations of states and phase transitions. II. Lattice gas and Ising model. Phys. Rev. 87, 410–419 (1952)

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

  37. Chang S.-C., Shrock R.: Zeros of Jones polynomials for families of knots and links. Phys. A 301, 196–218 (2001)

    Article  Google Scholar 

  38. X.-S. Lin, Zeros of the Jones polynomial, http://www.math.ucr.edu/~xl/abs-jk.pdf

  39. Champanerkar A., Kofman L.: On the Mahler measure of Jones polynomials under twisting. Algebr. Geom. Topol. 5, 1–22 (2005)

    Article  Google Scholar 

  40. Jin X., Zhang F.: Zeros of the Jones polynomials for families of pretzel links. Phys. A 328, 391–408 (2003)

    Article  Google Scholar 

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

    Article  Google Scholar 

  42. Bollobás B.: Modern Graph Theory. Springer, Berlin (1998)

    Book  Google Scholar 

  43. Adams C.C.: The Knot Book. American Mathematical Society, Providence, RI (2004)

    Google Scholar 

  44. Wang Y., Mueller J.E., Kemper B., Seeman N.C.: Assembly and characterization of five-arm and six-arm DNA branched junctions. Biochemistry 30, 5667–5674 (1991)

    Article  CAS  Google Scholar 

  45. Wang X., Seeman N.C.: Assembly and characterization of 8-arm and 12-arm DNA branched junctions. J. Am. Chem. Soc. 129, 8169–8176 (2007)

    Article  CAS  Google Scholar 

  46. Traldi L.: A dichroamtic polynomial for weighted graphs and link polynomials. Proc. Am. Math. Soc. 106, 279–286 (1989)

    Article  Google Scholar 

  47. Bondy J.A., Murty U.S.R.: Graph Theory and its Applications. Macmillan, London (1976)

    Google Scholar 

  48. Jones V.F.R.: A polynomial invariant for knots via Von Neumann algebras. Bull. Am. Math. Soc. 12, 103–112 (1985)

    Article  Google Scholar 

  49. Jones V.F.R.: Hecke algebra representations of braid groups and link polynomials. Ann. Math. 126, 335–388 (1987)

    Article  Google Scholar 

  50. Kauffman L.H.: State models and the Jones polynomial. Topology 26, 395–407 (1987)

    Article  Google Scholar 

  51. Kauffman L.H.: New invariants in the theory of knots. Am. Math. Mon. 95, 195–242 (1988)

    Article  Google Scholar 

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

    Article  Google Scholar 

  53. Tutte W.T.: A contribution to the theory of chromatic polynomials. Can. J. Math. 6, 80–91 (1954)

    Article  Google Scholar 

  54. Read R.C., Whitehead E.G. Jr: Chromatic polynomials of homeomorphism classes of graphs. Discrete Math. 204, 337–356 (1999)

    Article  Google Scholar 

  55. Shahmohamad H.: A survey on flow polynomial. Utilitas Mathematica 62, 13–32 (2002)

    Google Scholar 

  56. Murasugi K.: Knot Theory and its Applications. Birkhauser, Boston, MA (1996)

    Google Scholar 

  57. Read R.C.: Chain polynomials of graphs. Discrete Math. 265, 213–235 (2003)

    Article  Google Scholar 

  58. Harary F.: Graph Theory. Addison-Wesley, Reading, MA (1969)

    Google Scholar 

  59. Jin X., Zhang F.: On computing Kauffman bracket polynomial of Montesinos links. J. Knot Theory Ramif. 19(8), 1001–1023 (2010)

    Article  Google Scholar 

  60. Traldi L.: Chain polynomials and Tutte polynomials. Discrete Math. 248, 279–282 (2002)

    Article  Google Scholar 

  61. X. Jin, F. Zhang, The Homfly and dichromatic polynomials. Proc. Am. Math. Soc. (acceptted)

  62. He Y., Ye T., Su M., Zhang C., Ribbe A.E., Jiang W., Mao C.: Hierarchical self-assemly of DNA into symmetric supramolecular polyhedra. Nature 452, 198–203 (2008)

    Article  CAS  Google Scholar 

  63. Lin C., Liu Y., Yan H.: Designer DNA Nanoarchitectures. Biochemistry 48, 1663–1674 (2009)

    Article  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, Fujian, People’s Republic of China

    Xian’an Jin & Fuji Zhang

Authors
  1. Xian’an Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fuji Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xian’an Jin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jin, X., Zhang, F. The architecture and the Jones polynomial of polyhedral links. J Math Chem 49, 2063–2088 (2011). https://doi.org/10.1007/s10910-011-9876-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-011-9876-6

Keywords

Search

Navigation