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 TMF, 2003, Volume 134, Number 2, Pages 164–184 (Mi tmf155)

Topological Correlations in Trivial Knots: New Arguments in Favor of the Representation of a Crumpled Polymer Globule

O. A. Vasil'eva, S. K. Nechaevab

a L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences
b Paris-Sud University 11

Abstract: 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

DOI: https://doi.org/10.4213/tmf155

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English version:
Theoretical and Mathematical Physics, 2003, 134:2, 142–159

Bibliographic databases:

Citation: O. A. Vasil'ev, S. K. Nechaev, “Topological Correlations in Trivial Knots: New Arguments in Favor of the Representation of a Crumpled Polymer Globule”, TMF, 134:2 (2003), 164–184; Theoret. and Math. Phys., 134:2 (2003), 142–159

Citation in format AMSBIB
\Bibitem{VasNec03} \by O.~A.~Vasil'ev, S.~K.~Nechaev \paper Topological Correlations in Trivial Knots: New Arguments in Favor of the Representation of a Crumpled Polymer Globule \jour TMF \yr 2003 \vol 134 \issue 2 \pages 164--184 \mathnet{http://mi.mathnet.ru/tmf155} \crossref{https://doi.org/10.4213/tmf155} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2025792} \elib{http://elibrary.ru/item.asp?id=13436545} \transl \jour Theoret. and Math. Phys. \yr 2003 \vol 134 \issue 2 \pages 142--159 \crossref{https://doi.org/10.1023/A:1022267802220} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000181522200001} 

• http://mi.mathnet.ru/eng/tmf155
• https://doi.org/10.4213/tmf155
• http://mi.mathnet.ru/eng/tmf/v134/i2/p164

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Vasil'ev AN, “A multicomponent fluid system in the critical region in the presence of spatial confinement”, High Temperature, 42:4 (2004), 652–655
2. Vasil'ev AN, “Peculiarities of critical light opalescence in a spatially limited multicomponent system”, Optics and Spectroscopy, 98:6 (2005), 884–888
3. Mansfield M.L., “Development of knotting during the collapse transition of polymers”, The Journal of Chemical Physics, 127:24 (2007), 244902
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5. Athanasopoulou L., Athanasopoulos S., Karamanos K., Almirantis Ya., “Scaling properties and fractality in the distribution of coding segments in eukaryotic genomes revealed through a block entropy approach”, Phys Rev E, 82:5, Part 1 (2010), 051917
6. Klimopoulos A., Sellis D., Almirantis Ya., “Widespread Occurrence of Power-Law Distributions in Inter-Repeat Distances Shaped by Genome Dynamics”, Gene, 499:1 (2012), 88–98
7. Athanasopoulou L., Sellis D., Almirantis Ya., “A Study of Fractality and Long-Range Order in the Distribution of Transposable Elements in Eukaryotic Genomes Using the Scaling Properties of Block Entropy and Box-Counting”, Entropy, 16:4 (2014), 1860–1882
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