Abstract
This chapter reviews a few conceptual steps in an analytic description of topological interactions involving topology and statistical physics of fluctuating non-phantom rope-like objects. The main ingredient here is the statistics of Brownian bridges in a non-Euclidean space of constant negative curvature. After an introduction to the role of knots and topological constraints in polymeric systems, following topics are discussed, (i) the conformal methods for entangled random walks, (ii) conditional Brownian bridges in hyperbolic spaces, and (iii) the crumpled globule phase of polymers.
Access provided by CONRICYT-eBooks. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
L. H. Kauffman, Topology, 26 395 (1987); L.H. Kauffman, AMS Contemp. Math. Series, 78 263 (1989); L.H. Kauffman, Knots and Physics (Singapore: WSPC, 1991)
A. Yu. Grosberg, A. R. Khokhlov, Statistical physics of macromolecules (New York: AIP Press, 1994)
A. Yu. Grosberg, S. Nechaev, J. Phys. A: Math. Gen., 25 4659 (1992)
A. Yu. Grosberg, S. K. Nechaev, E. I. Shakhnovich, J.Phys. (Paris), 49 2095 (1988)
A. Yu. Grosberg, A. Feigel, Y. Rabin, Phys. Rev. E, 54, 6618 (1996)
E. Lieberman-Aiden, N. L. van Berkum, L. Williams, M. Imakaev, T. Ragoczy, A. Telling, I. Amit, B.R. Lajoie, P.J. Sabo, M. O. Dorschner, et al, Science 326 289 (2009)
S. F. Edwards, Proc. Roy. Soc., 91 513 (1967)
F. Wiegel, Introduction to Path-Integrals Methods in Physics and Polymer Science (Singapore: WSPC, 1986)
M. Yor, J. Appl. Prob., 29 202 (1992); Math. Finance, 3 231 (1993)
A. Grosberg, H. Frisch, J. Phys. A: Math. Gen., 37 3071 (2004)
E. Bogomolny Scattering on two Aharonov-Bohm vortices, arXiv:1508.04616
A. V. Vologodskii, M. D. Frank-Kamenetskii, Usp. Fiz. Nauk, 134 641 (1981) (in Russian); A. V. Vologodskii, A. V. Lukashin, M. D. Frank-Kamenetskii, V. V. Anshelevich, Zh. Exp. Teor. Fiz., 66 2153 (1974); A. V. Vologodskii, A. V. Lukashin, M.D. Frank-Kamenetskii, Zh. Exp. Teor. Fiz., 67 1875 (1974); M.D. Frank-Kamenetskii, A. V. Lukashin, A. V. Vologodskii, Nature, 258 398 (1975)
V. F. R. Jones, Ann. Math., 126, 335 (1987); Bull. Am. Math. Soc., 12 103 (1985)
W. B. R. Likorish, Bull. London Math. Soc., 20 558 (1988); W.B.R. Lickorish, An Introduction to Knot Theory, Springer Serirs: Graduate texts in Mathematics (Springer: 1997)
J. Birman, Knots, Links and Mapping Class Groups, Ann. Math. Studies, 82 (Princeton Univ. Press, 1976)
Integrable Models and Strings, Lect. Not. Phys., 436 (Springer: Heidelberg, 1994); M. Wadati, T. K. Deguchi, Y. Akutsu, Phys. Rep., 180 247 (1989)
L. H. Kauffman, H. Saleur, Comm. Math. Phys., 141 293 (1991)
V. A. Vassiliev, Complements of Discriminants of Smooth Maps: Topology and Applications in Math. Monographs, 98 (Trans. AMS, 1992); D. Bar-Natan, Topology, 34 423 (1995)
M. Khovanov, A categorification of of the Jones Polynomial, arXiv:math.QA/9908171; D. Bar-Natan, J. Knot Theory Ramifications, 16 243 (2007)
P. Tait, Trans. Royal Soc. Edinburgh 28 145 (1877)
P. Bangert, M. Berger, R. Prandi, J. Phys. (A): Math. Gen., 35 43 (2002)
S. R. Quake, Phys. Rev. Lett., 73 3317 (1994)
Y. Sheng, P. Lai, Phys. Rev. E, 63 021506 (2001)
E. Janse van Rensburg, D. Sumners, E. Wasserman, S. Whittington, J. Phys. A: Math. Gen, 25 6557 (1992)
C. Soteros, D. Sumners, S. Whittington, Math. Proc. Camb. Phil. Soc., 111, 75 (1992)
M. Freedman, Z. He, Z. Wang, Ann. Math. 139 1 (1994)
A. Kholodenko, D. Rolfsen, J. Phys. (A): 29 5677 (1996)
O. Karpenkov, A. B. Sossinsky, Russ. J. Math. Phys., 18 306 (2011), arXiv:1106.3414
Ideal Knots, eds. A. Stasiak, V. Katrich, L. H. Kauffman, Series of Knots and Everything, 19 (WSPC: Singapore, 1998)
A. Yu. Grosberg, S. Nechaev, Europhys. Lett., 20 603 (1992)
O. A. Vasilyev, S. K. Nechaev, JETP, 93 1119 (2001); O.A. Vasilyev, S.K. Nechaev, Theor. Math. Phys., 134 142 (2003)
P. G. de Gennes, Scaling Concepts in Polymer Physics (Cornell Univ. Press: Ithaca, 1979)
E. Helfand, D. Pearson, J. Chem. Phys., 79 2054 (1983); M. Rubinstein, E. Helfand, J. Chem. Phys., 82 2477 (1985)
A. Khokhlov, S. Nechaev, Phys. Lett. (A) 112 156 (1985)
S. Nechaev, Ya. G. Sinai, Bol. Soc. Bras. Mat., 21 121 (1991)
S. Nechaev, O. Vasilyev, J. Knot Theory Ramifications, 14 243 (2005); S. Nechaev, O. Vasilyev, Thermodynamics and topology of disordered knots: Correlations in trivial lattice knot diagrams, in Physical and Numerical Models in Knot Theory, chapter 22, 421, Series on Knots and Everything, (WSPC: Singapore, 2005)
S. K. Nechaev, A. N. Semenov, M. K. Koleva, Physica A, 140 506 (1987)
S. K. Nechaev, J. Phys. A: Math. Gen., 21 3659 (1988)
Li-Chien Shen, On Hecke groups, Schwarzian triangle functions and a class of hyper-elliptic functions, Ramanujan J. 39, 609 (2016)
W. Koppenfels, F. Stallman, Praxis der conformen abbildung, (Berlin: Springer, 1959)
C. Carathéodory, Functiontheorie II, (Basel, 1950)
V. V. Golubev, Lectures on Analytic Theory of Differential Equations (Moscow: GITTL, 1950)
E. Hille, Ordinary Differential Equations in the Complex Domain, (J. Willey & Sons: 1976)
K. Chandrassekharan, Elliptic Functions (Berlin: Springer, 1985)
M. Mumford, Tata Lectures on Theta, I, II, Progress in Mathematics 28, 34 (Boston, MA: Birkhauser, 1983)
H. Furstenberg, Trans. Am. Math. Soc. 198 377 (1963)
I. M. Lifshitz, A. Yu. Grosberg, JETP, 65 2399 (1973)
M. Tesi, E. Janse van Rensburg, E. Orlandini, S. Whittington, J. Phys. A: Math. Gen., 27 347 (1994)
E. Orlandini, M. Tesi, S. Whittington, J. Phys. (A): Math. Gen., 33 L-181 (2000)
M. Imakaev, K. Tchourine, S. Nechaev, L. Mirny, Soft Matter 11 (2015), 665
V. G. Rostiashvili, N.-K. Lee, T. A. Vilgis, J. Chem. Phys., 118 937 (2002)
C. F. Abrams, N.-K. Lee, S. Obukhov, Europhysics Letters, 59 391 (2002)
B. Chu, Q. Ying, A. Y. Grosberg, Macromolecules, 28 180 (1995)
A. Chertovich, P. Kos, J. Chem. Phys., 141 134903 (2014)
L. A. Mirny, Chromosome Research, 19 37 (2011)
R. D. Schram, G. T. Barkema, and H. Schiessel, J. Chem. Phys., 138 224901 (2013)
Y. Markaki, M. Gunkel, L. Schermelleh, S. Beichmanis, J. Neumann, M. Heidemann, H. Leonhardt, D. Eick, C. Cremer, T. Cremer, Cold Spring Harbor Symp. Quant. Biol., 75 475 (2010)
J. Dekker, M. A. Marti-Renom, L. A. Mirny, Nat. Rev. Genet., 14, 390 (2013)
A. Rosa and R. Everaers, PLoS Comput. Biol., 4 e1000153 (2008)
A. Y. Grosberg, Polym. Sci., Ser. C, 54 1 (2012);
T. Vettorel, A. Y. Grosberg, K. Kremer, Phys. Biol., 6 025013 (2009); J. D. Halverson, W. B. Lee, G. S. Grest, A. Y. Grosberg, K. Kremer, J. Chem. Phys., 134 204904 (2011)
J. Smrek, A. Y. Grosberg, Understanding the dynamics of rings in the melt in terms of annealed tree model, arXiv:1409.1483
J. Dorier, A. Stasiak, Nucleic Acids Res., 37 6316 (2009)
V. A. Avetisov, V. Ivanov, D. Meshkov S. Nechaev, JETP Lett., 98 242 (2013); V. A. Avetisov, V. A. Ivanov, D. A. Meshkov, S. K. Nechaev, Biophysical Journal, 107 2361 (2014)
J. D. Halverson, J. Smrek, K. Kremer, A. Y. Grosberg, Rep. Prog. Phys., 77 022601 (2014)
P. Virnau, L. A. Mirny, M. Kardar, PLoS Comput. Biol., 2 e122 (2006)
G. Kolesov, P. Virnau, M. Kardar and L. A. Mirny, Nucleic Acids Res., 35 W-425 (2007)
A. Khokhlov, S. Nechaev, J. de Physique II, 6 1547 (1996)
A. Yu. Grosberg, S. K. Nechaev, Macromolecules, 24 2789 (1991)
K. Urayama, S. Kohjiya, Polymer 38 955 (1997)
J. Ma, J. E. Straub, E. I. Shakhnovich, J. Chem. Phys., 103 2615 (1995)
R. Lua, A. L. Borovinskiy and A. Y. Grosberg, Polymer, 45 717 (2004)
J. Dekker, K. Rippe, M. Dekker, and N. Kleckner, Science 295 1306 (2002)
J. E. Dixon, S. Selvaraj, F. Yue, A. Kim, Y. Li, Y. Shen, M. Hu, J. S. Liu, and B. Ren, Nature 485 376 (2012)
T. Sexton, E. Yaffe, E. Kenigsberg, F. Bantignies, B. Leblanc, M. Hoichman, H. Parrinello, A. Tanay, G. Cavalli, Cell 148 458 (2012)
Y. Zhang, R. P. McCord, Y.-J. Ho, B. R. Lajoie, D. G. Hildebrand, A. C. Simon, M. S.Becker, F. W. Alt,J. Dekker, Cell 148 908 (2012)
S. Sofueva, E. Yaffe, W.-C. Chan, D. Georgopoulou, M. V. Rudan, H. Mira-Bontenbal, S. M. Pollard, G. P. Schroth, A. Tanay, S. Hadjur, The EMBO Journal, advance online publication (2013) doi:10.1038/emboj.2013.237
T. B. K. Le, M. V. Imakaev, L. A. Mirny, M. T. Laub, Science 342 731 (2013)
A. Yu. Grosberg, Soft Matter, 10, 560 (2014)
R. K. Sachs, G. van der Engh, B. Trask, H. Yokota, and J. E. Hearst, Proc. Nat. Acad. sci., 92, 2710 (1995)
C. Münkel, and J. Langowski, Phys. Rev. E, 57, 5888 (1998)
J. Ostashevsky, Mol. Biol. of the Cell, 9 3031 (1998)
J. Mateos-Langerak, M. Bohn, W. de Leeuw, O. Giromus, E. M. M. Manders, P. J. Verschure, M. H. G. Indemans, H. J. Gierman, D. W. Heerman, R. van Driel, and S. Goetze, Proc. Nat. Acad. Sci., 106, 3812 (2009)
B. V. S. Iyer, and G. Arya, Phys. Rev. E, 86, 011911 (2012)
M. Barbieri, M. Chotalia, J. Fraser, L.-M. Lavitas, J. Dostie, A. Pombo, and M. Nicodemi, Proc. Nat. Acad. Sci., 109, 16173 (2012)
C. C. Fritsch, J. Longowski, Chromosome Res., 19, 63 (2011)
A. Rosa, R. Everaers, Phys. Rev. Letters, 112, 118302 (2014)
M. Tamm, L. Nazarov, A. Gavrilov, and A. Chertovich, Phys. Rev. Lett., 114 178102 (2015)
V. A. Avetisov, L. Nazarov, S. K. Nechaev, M. V. Tamm, Soft Matter, 11 1019 (2015)
G. J. Filion, J. G. van Bemmel, U. Braunschweig, W. Talhout, J. Kind, L. D. Ward, W. Brugman, I. de Castro Genebra de Jesus, R. M. Kerkhoven, H. J. Bussemaker, B. van Steensel, Cell, 143, 212 (2010)
J. Desbois, S. Nechaev, 88 201 (1997); J. Phys. A: Math. Gen., 31 2767 (1998); A. Comtet, S. Nechaev, J. Phys. A: Math. Gen., 31 5609 (1998)
A. M. Vershik, S. Nechaev, R. Bikbov, Statistical properties of locally free groups with application to braid groups and growth of heaps, Comm. Math. Phys., 212 (2000), 469-501
N. Haug, S. Nechaev, M. Tamm, J. Stat. Mech. P-10013 (2014)
P. Serna, G. Bunin, A. Nahum, Phys. Rev. Lett., 115 228303 (2015)
A. Gorsky, S. Nechaev, R. Santachiara, G. Schehr, Nuclear Physics B 862 [FS] 167 (2012)
G. Bunin, M. Kardar, Coalescence Model for Crumpled Globules Formed in Polymer Collapse, Phys. Rev. Lett., 115 088303 (2015)
G. Palyi, C. Zucci, L. Caglioti (Eds.) Fundamentals of Life (Elsevier: Paris, 2002); Yu.N. Zhuravlev, V. A. Avetisov, Biogeosciences, 3 281 (2008); Yu. N. Zhuravlev, V. A. Avetisov, Hierarchical scale-free presentation of biological realm - Its Origin and Evolution in Biosphere Origin and Evolution, N. Dobretsov, N. Kolchanov A. Rozanov, G. Zavarzin (Eds.), (Springer: New York, 2008), 69; S. Wright, The roles of mutation, inbreeding, crossbreeding, and selection in evolution, Proceedings of the Sixth International Congress on Genetics 355 (1932); V. A. Avetisov, V.I. Goldanskii, PNAS 93 11435 (1996)
V. A. Avetisov, S. K. Nechaev, Geochemistry International, 52 1252 (2014)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer Nature Singapore Pte Ltd. and Hindustan Book Agency
About this chapter
Cite this chapter
Nechaev, S. (2017). Concepts of polymer statistical topology. In: Bhattacharjee, S., Mj, M., Bandyopadhyay, A. (eds) Topology and Condensed Matter Physics. Texts and Readings in Physical Sciences, vol 19. Springer, Singapore. https://doi.org/10.1007/978-981-10-6841-6_15
Download citation
DOI: https://doi.org/10.1007/978-981-10-6841-6_15
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-6840-9
Online ISBN: 978-981-10-6841-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)