Abstract
Long polymer chains in solution can be highly self-or mutually entangled. Several kinds of constraints can influence entanglement properties; among them a crucial role is played by geometrical constraints, which confine the chain(s) to restricted spaces. In this paper we investigate, using analytical and numerical techniques, the effects of different kinds of geometrical constraints on the topological entanglement complexity of polymer chain(s), i.e. on the knotting and linking probabilities.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
P.G. De Gennes, Tight knots, Macromolecules 17 (1984), 703–704.
S.F. Edwards, Statistical mechanics with topological constraints: I, Proc. Phys. Soc. 91 (1967), 513–519.
S.A. Wasserman and N.R. Cozzarelli, Biochemical topology: applications to DNA recombination and replication, Science 232 (1986), 951–960.
D.W. Sumners, Knot theory and DNA, in New Scientific Applications of Geometry and Topology, Proceedings of Symposia in Applied Mathematics 45, D.W. Sumners, ed., Am. Math. Soc. (1992), 39–72.
A.V. Vologodskii, A.V. Lukashin, M.D. Frank-Kamenetskii and V.V. Anshelevich, The knot probability in statistical mechanics of polymer chains, Soy. Phys.-JETP 39 (1974), 1059–1063.
J.P.J. Michels and F.W. Wiegel, On the topology of a polymer ring, Proc. Roy. Soc. A 403 (1986), 269–284.
E.J. Janse Van Rensburg and S.G. Whittington, The knot probability in lattice polygons, J. Phys. A: Math. Gen. 23 (1990), 3573–3590.
K. Koniaris and M. Muthukumar, Self-entanglement in ring polymers, J. Chem. Phys. 95 (1991), 2873–81.
D.W. Sumners and S.G. Whittington, Knots in self-avoiding walks, J. Phys. A: Math. Gen. 21 (1988), 1689–1694.
N. Pippenger, Knots in random walks, Disc. Appl. Math. 25 (1989), 273–278.
C.E. Soteros, D.W. Sumners and S.G. Whittington, Entanglement complexity of graphs in Z 3, Math. Proc. Camb. Phil. Soc. 111 (1992), 75–91.
E.J. Janse Van Rensburg, D.W. Sumners, E. Wasserman and S.G. Whittington, Entanglement complexity of self-avoiding walks, J. Phys. A: Math. Gen. 25 (1992), 6557–66.
A.V. Vologodskii, A.V. Lukashin and M.D. Frank-Kamenetskii, Topological interactions between polymer chains, Soy. Phys.-JETP 40 (1974), 932–936.
K.V. Klenin, A.V. Vologodskii, V.V. Anshelevich, A.M. Dykhne and M.D. Frank-Kamenetskii, Effect of excluded volume on topological properties of circular DNA, J. Biomol. Str. Dyn. 5 (1988), 1173–85.
W.F. Pohl, The probability of linking of random closed curves, Lectures Notes in Mathematics 894 (1981), 113–126.
B. Duplantier, Linking Numbers, Contacts and Mutual Inductances of a Random Set of Closed Curves,Comm. Math. Phys. 82 (1981), 41–68.
E. Orlandini, E.J. Janse Van Rensburg, M.C. Tesi and S.G. Whittington, Random linking of lattice polygons, J. Phys. A: Math. Gen. 27 (1994), 335–45.
J.M. Hammersley, The number of polygons on a lattice, Proc. Camb. Phil. Soc. 57 (1961), 516–523.
J.M. Hammersley, Percolation processes II. The connective constant, Proc. Camb. Phil. Soc. 53 (1957), 642–645.
H. Kesten, On the number of self-avoiding walks, J. Math. Phys. 4 (1963), 960–969.
Y. Diao, N. Pippenger and D.W. Sumners, On random knots, J. Knot Theory and Ramifications 3 (1994), 419–429.
Y. Diao, The knotting of equilateral polygons in R 3, J. Knot Theory and Ramifications 2 (1994), 413–425.
J.M. Hammersley and S.G. Whittington, Self-avoiding walks in wedges, J. Phys. A: Math. Gen. 18 (1985), 101–111.
C.E. Soteros and S.G. Whittington, Lattice models of branched polymers: effects of geometrical constraints, J. Phys. A: Math. Gen. 22 (1989), 5259–5270.
N. Madras and G. Slade, The Self-Avoiding Walk, Birkh5,user, Boston, 1993.
M.C. Tesi, E.J. Janse Van Rensburg, E. Orlandini and S.G. Whittington, Knot probability for lattice polygons in confined geometries, J. Phys. A: Math. Gen. 27 (1994), 347–60.
C.E. Soteros, Knots in graphs in subsets of Z 3, in this IMA volume.
D. Rolfsen, Knots and Links, Mathematics Lecture Series 7, Publish or Perish, Inc., Houston, Texas, 1976.
G. Torres, On the Alexander polynomial, Ann. Math. 57 (1953), 57–89.
M. Lal, “Monte Carlo” computer simulations of chain molecules. I., Molec. 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, 50, J. Stat. Phys. (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.
E.J. Janse Van Rensburg, S.G. Whittington and N. Madras, The pivot algorithm and polygons, J. Phys. A: Math. Gen. 23 (1990), 1589–1612.
D.E. Knuth, The Art of Computer Programming, volume 3, Addison-Wesley, Reading MA, 1973.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Tesi, M.C., van Rensburg, E.J.J., Orlandini, E., Whittington, S.G. (1998). Topological Entanglement Complexity of Polymer Chains in Confined Geometries. In: Whittington, S.G., De Sumners, W., Lodge, T. (eds) Topology and Geometry in Polymer Science. The IMA Volumes in Mathematics and its Applications, vol 103. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1712-1_11
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1712-1_11
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-98580-0
Online ISBN: 978-1-4612-1712-1
eBook Packages: Springer Book Archive