Abstract
This paper extends the knot polynomial classification of DNA knots and catenanes, by incorporating a measure of supercoiling. Cozzarelli, Millett, and White have used the Jones polynomial and the generalised 2-variable polynomial to describe the products of iterated Tn3 resolvase recombination and phage λ integrase mediated recombination. A new polynomial invariant, Ω, is introduced; based on the regular isotopy invariants of Kauffman. The Ω polynomial is an invariant of framed links and involves the Whitney degree of the link. This is useful because it not only allows a regular isotopy classification, but also distinguishes between plectonemic and solenoidal supercoils. The enzymes above require plectonemic supercoils for the synaptic substrate, so we put the Ω polynomial to use to investigate the supercoiling of the recombination products.
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References
Conway, J. H., An enumeration of knots and links and some of their algebraic properties, in E. Leech (ed.), Computational Problems in Abstract Algebra, Pergamon Press, Oxford, 1970, pp. 329–358.
Cozzarelli, N. R., Dungan, J. M., and Wasserman, S. A., Discovery of a predicted knot substantiates a model for site-specific recombination, Science 229 (1985), 171–174.
Cozzarelli, N. R., Millett, K. C., and White, J. H., Description of the topological entanglement of DNA knots and catenanes by a powerful method involving strand passage and recombination, J. Mol. Biol. 197 (1987), 585–604.
Cozzarelli, N. R. and Wasserman, S. A., Biochemical topology: Applications to DNA recombination and replication, Science 232 (1986), 951–960.
Franks, J. and Williams, R. F., Braids and the Jones polynomial, Trans. Amer. Math. Soc. 303 (1987), 97–108.
Freyd, P., Hoste, J., Lickorish, W. B. R., Millett, K. C., Ocneanu, A., and Yetter, D., A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. 12 (1985), 239–246.
Fuller, F. B., The writhing number of a space curve, Proc. Natl. Acad. Sci. U.S.A. 68 (1971), 815–819.
Jones, V. F. R., A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. 12 (1985), 103–111.
Kauffman, L. H., On Knots, Princeton University Press, New Jersey, 1988.
Kaufman, L. H., An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), 417–471.
Koller, T. and Stasiak, A., Analysis of DNA knots and catenanes allows to deduce the mechanism of action of enzymes which cut and join DNA strands, in A. Amann, L. Cederbaum, W. Gans (eds), Fractals, Quasicrystals, Chaos, Knots and Algebraic Quantum Mechanics, Kluwer Acad. Publ., Dordrecht, 1988, pp. 207–219.
Lickorish, W. B. R. and Millett, K. C., A polynomial invariant of oriented links, Topology 26 (1987), 107–141.
Nash, H. A. and Pollock, T. J., Site-specific recombination of Bacteriophage Lambda, J. Mol. Biol. 170 (1983), 19–38.
NSF Centre for Mathematics and Molecular Biology, Project Summary and Proposals for Research, University of California, Berkeley Report (1988).
Rolfsen, D., Knots and Links, Publish or Perish, Berkeley, 1976.
Strogatz, S., Estimating the torsional rigidity of DNA from supercoiling data, J. Chem. Phys. 77 (1982), 580–581.
Sumners, D. W., The role of knot theory in DNA research, in Geometry and Topology, Marcel Dekker, New York, 1987, pp. 297–318.
Trace, B., On the Reidemeister moves of a classical knot, Proc. Amer. Math. Soc. 89 (1983), 722–724.
Whitney, H., On regular closed curves in the plane, Compositio Math. 4 (1937), 276–284.
White, J. H., An introduction to the geometry and topology of DNA structure, in Mathematical Methods for DNA Sequences, CRC Boca Raton, Florida, 1989, pp. 225–253.
Wilkinson, S. A., Applications of geometry and knot theory to DNA supercoiling and recombinations, Ph.D. Thesis, University of Lancaster (1990), submitted.
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Wilkinson, S.A. Modelling supercoiled DNA knots and catenanes by means of a new regular isotopy invariant. Acta Appl Math 25, 1–20 (1991). https://doi.org/10.1007/BF00047663
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DOI: https://doi.org/10.1007/BF00047663