Abstract
The torsion of self-avoiding polygons (SAPs) on the cubic lattice is studied by means of rigorous results and Monte Carlo simulations. More precisely, rigorous results can be obtained in the asymptotic limit of infinitely long chains, while numerical simulations are essential to investigate the behavior of chains of finite length. The relation between torsion and the geometrical entanglement of SAPs is explored, and the results obtained are compared with ones concerning the writhe, which is another measure of geometrical entanglement.
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References
P.G. De GennesTight knots, Macromolecules 17 (1984), 703–704.
S.F. EdwardsStatistical mechanics with topological constraints: I, Proc. Phys. Soc. 91 (1967), 513.
S.A. Wasserman, J.M. Dungan and N.R. CozzarelliDiscovery of a predicted DNA knot substantiates a model for site-specific recombination, Science 229 (1985), 171–174.
E.J. Janse Van Rensburg, E. OrlandinI, D.W. Sumners, M.C. Tesi and S.G. Whittington, Topology and geometry of biopolymers, in Mathematical approaches to bimolecular structure and Dynamics, IMA volume 82 (to appear).
W.R. Bauer, F.H.C. Crick and J.H. WhiteSupercoiled DNA, Sci. American 243 (1980), 118–133.
D.W. Sumners and S.G. WhittingtonKnots in self-avoiding walks, J. Phys. A: Math. Gen. 21 (1988), 1689–1694.
N. PippengerKnots in random walks Disc. Appl. Math. 25 (1989), 273–278.
C.S. Soteros, D.W. Sumners and S.G. WhittingtonEntanglement complexity of graphs in Z 3, Math. Proc. Camb. Phil. Soc. 111 (1992), 75–91.
A.V. Vologodskii, A.V. Lukashin, M.D. Frank-kamenetskii and V.V. AnshelevichThe knot probability in statistical mechanics of polymer chains, Sov. Phys-JEPT 39 (1974), 1059–1063.
E.J. Janse Van Rensburg and S.G. WhittingtonThe knot probability in lattice polygons, J. Phys. A: Math. Gen. 23 (1990), 3573–3590.
M.C. Tesi, E.J. Janse Van Rensburg, E. OrlandinI, D.W. Sumners and S.G. WhittingtonKnotting and supercoiling in circular DNA: a model incorporating the effect of added salt, Phys. Rev. E 49 (1994), 868–872.
E.J. Janse VaN Rensburg, E. ORlandini, D.W. Sumners, M.C. Tesi and S.G. WhittingtonThe writhe of a self-avoiding polygon,J. Phys. A: Math. Gen.26 (1993), L981–L986.
E. Orlandini, M.C. Tesi, S.G. Whittington, D.W. Sumners and E.J. Janse Van Rensburg, The writhe of a self-avoiding walk, J. Phys. A: Math. Gen.27 (1994), L333—L338.
D.J. Struik, Lectures on classical differential geometry, Dover Publications, NY, 1988, 232.
M.C. Tesi, E.J. Janse Van Rensburg, E. Orlandini and S.G. Whittington, Torsion of a self-avoiding polygon in three dimension J. Phys. A: Math. Gen. (1997).
F.B. Fuller, The Writhing Number of a Space Curve, Proc. Natl. Acad. Sci. USA 68 (1971), 815–819.
Y. Diao, N. Pippenger and D.W. Sumners, On random knots, J. Knot Theory and Ramifications 3 (1994), 419–429.
H. Kesten, On the number of self-avoiding walks, J. Math. Phys. 4 (1963), 960–969.
R.C. Lacher and D.W. Sumners, Data structures and algorithms for the computation of topological invariants of entanglements: link, twist and writhe,in Computer simulations of polymers, R.J Roe ed., Prentice-Hall (1991), 365–373.
M. Lal, “Monte Carlo” computer simulations of chain molecules. I, Mol. Phys. 17 (1969), 57–64.
N. Madras and A.D. Sokal, The pivot algorithm: a highly efficient Monte Carlo method for the self-avoiding walk, J. Stat. Phys. 50 (1988), 109–186.
N. Madras, A. Orlitsky and L.A. Shepp, Monte Carlo generation of self-avoiding walks with fixed endpoints and fixed length, J. Stat. Phys. 58 (1990), 159–183.
M.G. Bickis, The torsion of three-dimensional random walk, private communication.
C.C. Adams, The knot book, W.H. Freeman and Company, 1994, 306.
B. Berg and D. Foester, Random paths and random surfaces on a digital computer, Phys. Lett 106B (1981), 323–326.
C. Aragao De Carvalho, S. Caracciolo and J. Fröhlich, Polymers and 0)I 4 -theory in four dimensions Nucl. Phys. B [FS7] 215 (1983), 209–248.
E.J. Janse Van Rensburg and S.G. Whittington, The BFACF algorithm and knotted polygons,J. Phys. A: Math. Gen. 24 (1991), 5553–5567.
A.D. Sokal and L.E. Thomas, Exponential convergence to equilibrium for a class of random walk models, J. Stat. Phys. 54 (1989), 797–828.
E. Orlandini, Monte Carlo study of polymer systems by Multiple Markov Chain method, in this volume.
M.C. Tesi, E.J. Janse Van Rensburg, E. Orlandini and S.G. Whittington, Monte Carlo study of the interacting self-avoiding walk model in three dimensions, J. Stat. Phys. 82 (1996), 155–181.
D.M. Walba, Topological stereochemistry, Tetrahedron 41(16) (1985), 3161–3212.
E.J. Janse Van Rensburg, E. Orlandini, D.W. Sumners, M.C. Tesi and S.G. Whittington, The writhe of knots in the cubic lattice, J. Knot Theory and Ramifications, volume 6, 31–44, 1997.
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Tesi, M.C. (1998). Geometrical Entanglement in Lattice Models of Ring Polymers: Torsion and Writhe. In: Whittington, S.G. (eds) Numerical Methods for Polymeric Systems. The IMA Volumes in Mathematics and its Applications, vol 102. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1704-6_6
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