Advertisement

From Tangle Fractions to DNA

  • L. H. Kauffman
  • S. Lambropoulou
Part of the Biological and Medical Physics, Biomedical Engineering book series (BIOMEDICAL)

Keywords

Continue Fraction Lens Space Jones Polynomial Vertex Weight Reidemeister Move 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. M. Asaeda, J. H. Przytycki, A. S. Sikora, Kauffman-Harary conjecture holds for Montesinos knots. J. Knot Theory Ramifications 13 (4), 467-477 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    C. Bankwitz, H. G. Schumann, Abh. Math. Sem. Univ. Hamburg 10, 263-284 (1934)CrossRefGoogle Scholar
  3. 3.
    S. A. Bleiler, Y. H. Moriah, Splittings and branched coverings of B 3 , Math. Ann. 281(4),531-543 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    J. H. Conway, An enumeration of knots and links and some of their alge-braic properties, Proceedings of the Conference on Computational Problems in Abstract Algebra Held at Oxford in 1967, J. Leech ed. , (First edition 1970), (Pergamon Press, 1970), pp. 329-358Google Scholar
  5. 5.
    L. Siebenmann, Lecture Notes on Rational Tangles, Orsay (1972) (unpublished)Google Scholar
  6. 6.
    G. Burde, H. Zieschang, “Knots”, de Gruyter Studies in Mathematics 5 (1985)Google Scholar
  7. 7.
    L. H. Kauffman, Knot Logic, Knots and Applications, Series on Knots and Every-thing, 2, L. H. Kauffman ed. , World Scientific, (1995)Google Scholar
  8. 8.
    J. R. Goldman, L. H. Kauffman, Adv. Appl. Math. 18, 300-332 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    A. Kawauchi, ‘A Survey of Knot Theory ’ (Birkhäuser, Verlag, 1996)Google Scholar
  10. 10.
    W. B. R. Lickorish, ‘An Introduction to Knot Theory ’, Springer Graduate Texts in Mathematics 175 (1997)Google Scholar
  11. 11.
    K. Murasugi, ‘Knot Theory and its Applications ’, Translated from the 1993 Japanese original by B. Kurpita, Birkhäuser Verlag (1996)Google Scholar
  12. 12.
    L. H. Kauffman, Topology, 26, 395-407 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    L. H. Kauffman, S. Lambropoulou, On the classification of rational tangles, to ap-pear in Advances in Applied Math. (See http://www.math.uic.edu/˜kauffman/ or http://users.ntua.gr/sofial or math.GT/0311499)
  14. 14.
    H. Schubert, Math. Zeitschrift 65, 133-170 (1956)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    K. Reidemeister, Abh. Math. Sem. Hansischen Univ. 11, 102-109 (1936)CrossRefGoogle Scholar
  16. 16.
    G. Burde, Math. Zeitschrift 145, 235-242 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    L. H. Kauffman, S. Lambropoulou, On the classification of rational knots, to appear in L’ Enseignement Math. (See http://www.math.uic.edu/˜kauffman/or http://users.ntua.gr/sofial or math.GT/0212011)
  18. 18.
    K. Reidemeister, “Knotentheorie” (Reprint), Chelsea, New York (1948)Google Scholar
  19. 19.
    P. G. Tait, On knots, I, II, III, Scientific Papers, 1, Cambridge University Press, Cambridge, 273-347 (1898)Google Scholar
  20. 20.
    W. Menasco, M. Thistlethwaite, Annals of Mathematics 138, 113-171 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    A. Ya. Khinchin, ‘Continued Fractions ’, Dover (republication of the 1964 edition of Chicago Univ. Press) (1997)Google Scholar
  22. 22.
    C. D. Olds, ‘Continued Fractions ’, New Mathematical Library, Math. Assoc. of Amerika, 9 (1963)Google Scholar
  23. 23.
    K. Kolden, Arch. Math. og Naturvidenskab 6, 141-196 (1949)MathSciNetGoogle Scholar
  24. 24.
    L. H. Kauffman, Ann. Math. Stud. 115 (Princeton Univ. Press, Princeton, NJ, 1987)Google Scholar
  25. 25.
    L. H. Kauffman, S. Lambropoulou, On the classififcation of rational tangles. Adv. in Appln. Math. 33 (2), 199-237 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    L. H. Kauffman, S. Lambropoulou, On the classififcation of rational knots. Enseign. Math. (2) 49 (3-4) 357-410 (2003)zbMATHMathSciNetGoogle Scholar
  27. 27.
    V. F. R. Jones, Bull. Am. Math. Soc. (N. S. ) 12 (1), 103-111 (1985)zbMATHCrossRefGoogle Scholar
  28. 28.
    V. F. R. Jones, Notices Am. Math. Soc. 33 (2), 219-225 (1986)Google Scholar
  29. 29.
    D. A. Krebes, J. Knot Theory Ramifications 8 (3), 321-352 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    L. H. Kauffman, “Formal Knot Theory ”, Mathematical Notes 30, Princeton Univ. Press, Princeton, NJ, (1983)Google Scholar
  31. 31.
    L. Person, M. Dunne, J. DeNinno, B. Guntel and L. Smith, Colourings of ratio-nal, alternating knots and links, (preprint 2002)Google Scholar
  32. 32.
    L. H. Kauffman, F. Harary, Knots and Graphs I - Arc Graphs and Colourings, Advances in Applied Mathematics 22, 312-337 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    M. Asaeda, J. Przytycki, A. Sikora, Kauffman-Harary Conjecture holds for Montesinos Knots (to appear in JKTR)Google Scholar
  34. 34.
    W. Franz, J. Reine Angew. Math. 173, 245-254 (1935)zbMATHGoogle Scholar
  35. 35.
    C. Ernst, D. W. Sumners, Math. Proc. Camb. Phil. Soc. 102, 303-315 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    C. Ernst, D. W. Sumners, Math. Proc. Camb. Phil. Soc. 108, 489-515 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    D. W. Sumners, Math. Intelligencer 12, 71-80 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    N. Cozzarelli, F. Dean, T. Koller, M. A. Krasnow, S. J. Spengler and A. tasiak Nature 304, 550-560 (1983)Google Scholar
  39. 39.
    M. C. Culler, C. M. Gordon, J. Luecke and P. B. Shalen, Ann. Math. 125, 237-300 (1987)CrossRefMathSciNetGoogle Scholar
  40. 40.
    C. Ernst, D. W. Sumners, Math. Proc. Cambridge Philos. Soc. 126 (1), 23-36 (1999)zbMATHCrossRefMathSciNetADSGoogle Scholar
  41. 41.
    I. K. Darcy, Solving unoriented tangle equations involving 4-plats. J. Knot Theory Ramifications 14 (8), 993-1005 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    E. J. Brody, Ann. Math. 71, 163-184 (1960)CrossRefMathSciNetGoogle Scholar
  43. 43.
    J. M. Montesinos, Revetements ramifies des noeuds, Espaces fibres de Seifert et scindements de Heegaard, Publicaciones del Seminario Mathematico Garcia de Galdeano, Serie II, Seccion 3 (1984)Google Scholar
  44. 44.
    V. V. Prasolov, A. B. Sossinsky, “Knots, Links, Braids and 3-Manifolds”, AMS Translations of Mathematical Monographs 154 (1997)Google Scholar
  45. 45.
    K. Reidemeister Abh. Math. Sem. Univ. Hamburg 5, 24-32 (1927)CrossRefGoogle Scholar
  46. 46.
    D. Rolfsen, ‘Knots and Links ’ (Publish or Perish Press, Berkeley 1976)Google Scholar
  47. 47.
    H. Seifert, Abh. Math. Sem. Univ. Hamburg, 11, 84-101 (1936)CrossRefGoogle Scholar
  48. 48.
    J. Sawollek, Tait's flyping conjecture for 4-regular graphs, preprint (1998)Google Scholar
  49. 49.
    C. Sundberg, M. Thistlethwaite, The rate of growth of the number of alternating links and tangles, Pacific J. Math. 182(2), 329-358 (1998)zbMATHMathSciNetGoogle Scholar
  50. 50.
    H. S. Wall, ‘Analytic Theory of Continued Fractions ’ (D. Van Nostrand Com-pany, Inc. , 1948)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • L. H. Kauffman
    • 1
  • S. Lambropoulou
    • 2
  1. 1.Department of Mathematics, Statistics and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA
  2. 2.Department of MathematicsNational Technical University of AthensAthensGreece

Personalised recommendations