Geodesic
Topological Correlations in Trivial Knots: New Arguments in Favor of the Representation of a Crumpled Polymer Globule
Teoretičeskaâ i matematičeskaâ fizika, Tome 134 (2003) no. 2, pp. 164-184 Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove the representation of a fractal crumpled structure of a strongly collapsed unknotted polymer chain. In this representation, topological considerations result in the chain developing a densely packed system of folds, which are mutually segregated at all scales. We investigate topological correlations in randomly generated knots on rectangular lattices (strips) of fixed widths. We find the probability of the spontaneous formation of a trivial knot and the probability that each finite part of a trivial knot becomes a trivial knot itself after joining its ends in a natural way. The complexity of a knot is characterized by the highest degree of the Jones–Kauffman polynomial topological invariant. We show that the knot complexity is proportional to the strip length in the case of long strips. Simultaneously, the typical complexity of a “quasi-knot”, which is a part of a trivial knot, is substantially less. Our analysis shows that the latter complexity is proportional to the square root of the strip length. The results obtained clearly indicate that the topological state of any part of a trivial knot densely filling the lattice is also close to the trivial state.
Keywords: knots, polymers, topological invariants, Brownian bridge, non-Euclidean geometry. 
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     title = {Topological {Correlations} in {Trivial} {Knots:} {New} {Arguments} in {Favor} of the {Representation} of a {Crumpled} {Polymer} {Globule}},
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O. A. Vasil'ev; S. K. Nechaev. Topological Correlations in Trivial Knots: New Arguments in Favor of the Representation of a Crumpled Polymer Globule. Teoretičeskaâ i matematičeskaâ fizika, Tome 134 (2003) no. 2, pp. 164-184. http://geodesic.mathdoc.fr/item/TMF_2003_134_2_a1/

[1] I. M. Lifshits, ZhETF, 55 (1968), 2408

[2] I. M. Lifshits, A. Yu. Grosberg, A. R. Khokhlov, Rev. Mod. Phys., 50 (1978), 683 | DOI | MR

[3] A. Yu. Grosberg, A. R. Khokhlov, Statisticheskaya fizika makromolekul, Nauka, M., 1989 | MR

[4] A. Yu. Grosberg, S. K. Nechaev, E. I. Shakhnovich, J. Phys. (Paris), 49 (1988), 2095 | DOI

[5] B. B. Mandelbrot, The Fractal Geometry of Nature, Freeeman, San Francisco, 1982 | MR | Zbl

[6] B. Chu, Q. Ying, A. Grosberg, Macromolecules, 28 (1995), 180 | DOI

[7] A. R. Khokhlov, S. K. Nechaev, J. Phys. II (Paris), 6 (1996), 1547 | MR

[8] A. Yu. Grosberg, S. K. Nechaev, Macromolecules, 24 (1991), 2789 | DOI

[9] J. Ma, J. E. Straub, E. I. Shakhnovich, J. Chem. Phys., 103 (1995), 2615 | DOI

[10] V. F. R. Jones, Bull. Am. Math. Soc., 12 (1985), 103 | DOI | MR | Zbl

[11] L. H. Kauffman, Topology, 26 (1987), 395 | DOI | MR | Zbl

[12] F. Y. Wu, J. Knot. Theory Ramific., 1 (1992), 47 | DOI

[13] A. Yu. Grosberg, S. Nechaev, J. Phys. A, 25 (1992), 4659 | DOI | MR | Zbl

[14] O. A. Vasilev, S. K. Nechaev, ZhETF, 120 (2001), 1288

[15] I. M. Lifshits, A. Yu. Grosberg, ZhETF, 65 (1973), 2399

[16] L. B. Koralov, S. K. Nechaev, Ya. G. Sinai, Teoriya veroyatn. i ee primen., 38 (1993), 331 | MR | Zbl

[17] P. Bougerol, Probab. Theory Relat. Fields, 78 (1988), 193 | DOI | MR | Zbl

[18] S. Nechaev, Ya. G. Sinai, Bol. Soc. Bras. Mat. Nova Sér., 21 (1991), 121 | DOI | MR | Zbl

[19] A. V. Letchikov, UMN, 51:1 (1996), 51 | DOI | MR | Zbl

[20] A. Comtet, S. Nechaev, R. Voituriez, J. Stat. Phys., 102 (2001), 203 | DOI | MR | Zbl

[21] M. E. Gertsenshtein, V. B. Vasilev, Teoriya veroyatn. i ee primen., 4 (1959), 424

[22] H. Fürstenberg, Trans. Am. Math. Soc., 198 (1963), 377 ; В. Н. Тутубалин, Теория вероятн. и ее примен., 10 (1965), 19 ; 13 (1968), 63 | DOI | MR | MR | Zbl | MR | Zbl

[23] A. Yu. Grosberg, S. K. Nechaev, Adv. Polym. Sci., 106 (1993), 1 ; S. K. Nechaev, Statistics of Knots and Entangled Random Walks, World Scientific, Singapore, 1996 | DOI | MR | Zbl