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Tangles, Rational Knots and DNA

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Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1973)

Abstract

This paper draws a line from the basics of rational tangles to the tangle model of DNA recombination. We sketch the classification of rational tangles, unoriented and oriented rational knots and the application of these subjects to DNA recombination.

Keywords

  • Continue Fraction
  • Lens Space
  • Jones Polynomial
  • Reidemeister Move
  • Link Diagram

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.

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References

  1. M. Asaeda, J. Przytycki and A. Sikora, Kauffman-Harary conjecture holds for Montesinos knots, J. Knot Theory Ramifications 13 (2004), no. 4, 467–477.

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. C. Bankwitz and H.G. Schumann, Über Viergeflechte, Abh. Math. Sem. Univ. Hamburg, 10 (1934), 263–284.

    CrossRef  Google Scholar 

  3. S. Bleiler and J. Moriah, Heegaard splittings and branched coverings of B 3, Math. Ann., 281, 531–543.

    Google Scholar 

  4. E.J. Brody, The topological classification of the lens spaces, Annals of Mathematics, 71 (1960), 163–184.

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. G. Burde, Verschlingungsinvarianten von Knoten und Verkettungen mit zwei Brücken, Math. Zeitschrift, 145 (1975), 235–242.

    CrossRef  MATH  MathSciNet  Google Scholar 

  6. G. Burde, H. Zieschang, “Knots”, de Gruyter Studies in Mathematics 5 (1985).

    Google Scholar 

  7. J.H. Conway, An enumeration of knots and links and some of their algebraic properties, Proceedings of the conference on Computational problems in Abstract Algebra held at Oxford in 1967, J. Leech ed., (First edition 1970), Pergamon Press, 329–358.

    Google Scholar 

  8. N. Cozzarelli, F. Dean, T. Koller, M. A. Krasnow, S.J. Spengler and A. Stasiak, Determination of the absolute handedness of knots and catenanes of DNA, Nature, 304 (1983), 550–560.

    Google Scholar 

  9. M.C. Culler, C.M. Gordon, J. Luecke and P.B. Shalen, Dehn surgery on knots, Annals of Math., 125 (1987), 237–300.

    CrossRef  MATH  MathSciNet  Google Scholar 

  10. I. Darcy, Solving oriented tangle equations involving 4-plats, J. Knot Theory Ramifications 14 (2005), no. 8, 1007–1027.

    CrossRef  MATH  MathSciNet  Google Scholar 

  11. I. Darcy, Solving unoriented tangle equations involving 4-plats, J. Knot Theory Ramifications 14 (2005), no. 8, 993–1005.

    CrossRef  MATH  MathSciNet  Google Scholar 

  12. C.Ernst, D.W. Sumners, The growth of the number of prime knots, Math. Proc. Camb. Phil. Soc., 102 (1987), 303–315.

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. C.Ernst, D.W. Sumners, A calculus for rational tangles: Applications to DNA Recombination, Math. Proc. Camb. Phil. Soc., 108 (1990), 489–515.

    CrossRef  MATH  MathSciNet  Google Scholar 

  14. C. Ernst, D. W. Sumners, Solving tangle equations arising in a DNA recombination model. Math. Proc. Cambridge Philos. Soc., 126, No. 1 (1999), 23–36.

    CrossRef  ADS  MATH  MathSciNet  Google Scholar 

  15. W. Franz, Über die Torsion einer Überdeckung, J. Reine Angew. Math. 173 (1935), 245–254.

    Google Scholar 

  16. J.R. Goldman, L.H. Kauffman, Knots, Tangles and Electrical Networks, Advances in Applied Math., 14 (1993), 267–306.

    CrossRef  MATH  MathSciNet  Google Scholar 

  17. J.R. Goldman, L.H. Kauffman, Rational Tangles, Advances in Applied Math., 18 (1997), 300–332.

    CrossRef  MATH  MathSciNet  Google Scholar 

  18. V.F.R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985) no. 1, 103–111.

    CrossRef  MATH  MathSciNet  Google Scholar 

  19. V.F.R. Jones, A new knot polynomial and von Neumann algebras, Notices Amer. Math. Soc. 33 (1986), no. 2, 219–225.

    MathSciNet  Google Scholar 

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

    CrossRef  MATH  MathSciNet  Google Scholar 

  21. L.H. Kauffman, An invariant of regular isotopy. Transactions of the Amer. Math. Soc., 318 (1990), No 2, 417–471.

    CrossRef  MATH  MathSciNet  Google Scholar 

  22. L.H. Kauffman, Knot Logic, Knots and Applications, Series on Knots and Everything, 2, L.H. Kauffman ed., World Scientific, (1995).

    Google Scholar 

  23. L.H. Kauffman, “On knots”, Ann. of Math. Stud. 115, Princeton Univ. Press, Princeton, N.J., (1987).

    Google Scholar 

  24. L.H. Kauffman, “Formal Knot Theory”, Mathematical Notes 30, Princeton Univ. Press, Princeton, N.J., (1983), republished by Dover Publications (2006).

    Google Scholar 

  25. L.H. Kauffman, F. Harary, Knots and Graphs I - Arc Graphs and Colorings, Advances in Applied Mathematics, 22 (1999), 312–337.

    CrossRef  MATH  MathSciNet  Google Scholar 

  26. L.H. Kauffman, S. Lambropoulou, On the classification of rational tangles, Advances in Applied Math. 33, No. 2 (2004), 199–237.

    CrossRef  MATH  MathSciNet  Google Scholar 

  27. L.H. Kauffman, S. Lambropoulou, On the classification of rational knots, L' Enseignement Mathematiques 49 (2003), 357–410.

    MATH  MathSciNet  Google Scholar 

  28. A. Kawauchi, “A Survey of Knot Theory”, Birkhäuser Verlag (1996).

    Google Scholar 

  29. A.Ya. Khinchin, “Continued Fractions”, Dover (1997) (republication of the 1964 edition of Chicago Univ. Press).

    Google Scholar 

  30. K. Kolden, Continued fractions and linear substitutions, Archiv for Math. og Naturvidenskab, 6 (1949), 141–196.

    MathSciNet  Google Scholar 

  31. D.A. Krebes, An obstruction to embedding 4-tangles in links. J. Knot Theory Ramifications 8 (1999), no. 3, 321–352.

    CrossRef  MATH  MathSciNet  Google Scholar 

  32. W.B.R. Lickorish, “An introduction to knot theory”, Springer Graduate Texts in Mathematics, 175 (1997).

    Google Scholar 

  33. W. Menasco, M. Thistlethwaite, The classification of alternating links, Annals of Mathematics, 138 (1993), 113–171.

    CrossRef  MATH  MathSciNet  Google Scholar 

  34. 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 

  35. K. Murasugi, “Knot Theory and Its Applications”, Translated from the 1993 Japanese original by Bohdan Kurpita. Birkhuser Boston, Inc., Boston, MA, (1996). viii+341 pp.

    Google Scholar 

  36. C.D. Olds, “Continued Fractions”, New Mathematical Library, Math. Assoc. of Amerika, 9 (1963).

    Google Scholar 

  37. L. Person, M. Dunne, J. DeNinno, B. Guntel and L. Smith, Colorings of rational, alternating knots and links, (preprint 2002, unpublished).

    Google Scholar 

  38. V.V. Prasolov, A.B. Sossinsky, “Knots, Links, Braids and 3-Manifolds”, AMS Translations of Mathematical Monographs 154 (1997).

    Google Scholar 

  39. K. Reidemeister, Elementare Begründung der Knotentheorie, Abh. Math. Sem. Univ. Hamburg, 5 (1927), 24–32.

    CrossRef  Google Scholar 

  40. K. Reidemeister, “Knotentheorie” (Reprint), Chelsea, New York (1948).

    Google Scholar 

  41. K. Reidemeister, Knoten und Verkettungen, Math. Zeitschrift, 29 (1929), 713–729.

    CrossRef  MATH  MathSciNet  Google Scholar 

  42. K. Reidemeister, Homotopieringe und Linsenräume, Abh. Math. Sem. Hansischen Univ., 11 (1936), 102–109.

    CrossRef  Google Scholar 

  43. D. Rolfsen, “Knots and Links”, Publish or Perish Press, Berkeley (1976).

    Google Scholar 

  44. H. Seifert, Die verschlingungsinvarianten der zyklischen knotenüberlagerungen, Abh. Math. Sem. Univ. Hamburg, 11 (1936), 84–101.

    CrossRef  Google Scholar 

  45. J. Sawollek, Tait's flyping conjecture for 4-regular graphs, J. Combin. Theory Ser. B 95 (2005), no. 2, 318–332.

    CrossRef  MATH  MathSciNet  Google Scholar 

  46. H. Schubert, Knoten mit zwei Brücken, Math. Zeitschrift, 65 (1956), 133–170.

    CrossRef  MATH  MathSciNet  Google Scholar 

  47. L. Siebenmann, Lecture Notes on Rational Tangles, Orsay (1972) (unpublished).

    Google Scholar 

  48. D.W. Sumners, Untangling DNA, Math.Intelligencer, 12 (1990), 71–80.

    CrossRef  MATH  MathSciNet  Google Scholar 

  49. C. Sundberg, M. Thistlethwaite, The rate of growth of the number of alternating links and tangles, Pacific J. Math., 182 No. 2 (1998), 329–358.

    CrossRef  MATH  MathSciNet  Google Scholar 

  50. P.G. Tait, On knots, I, II, III, Scientific Papers, 1 (1898), Cambridge University Press, Cambridge, 273–347.

    Google Scholar 

  51. H.S. Wall, “Analytic Theory of Continued Fractions”, D. Van Nostrand Company, Inc. (1948).

    Google Scholar 

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Correspondence to Louis H. Kauffman or Sofia Lambropoulou .

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Kauffman, L.H., Lambropoulou, S. (2009). Tangles, Rational Knots and DNA. In: Ricca, R. (eds) Lectures on Topological Fluid Mechanics. Lecture Notes in Mathematics(), vol 1973. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00837-5_3

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