Topological correlations in trivial knots: new arguments in support of the crumpled polymer globule
Abstract
We prove the fractal crumpled structure of collapsed unknotted polymer ring. In this state the polymer chain forms a system of densely packed folds, mutually separated in all scales. The proof is based on the numerical and analytical investigation of topological correlations in randomly generated dense knots on strips $L_{v} \times L_{h}$ of widths $L_{v}=3,5$. We have analyzed the conditional probability of the fact that a part of an unknotted chain is also almost unknotted. The complexity of dense knots and quasi--knots is characterized by the power $n$ of the Jones--Kauffman polynomial invariant. It is shown, that for long strips $L_{h} \gg L_{v}$ the knot complexity $n$ is proportional to the length of the strip $L_{h}$. At the same time, the typical complexity of the quasi--knot which is a part of trivial knot behaves as $n\sim \sqrt{L_{h}}$ and hence is significantly smaller. Obtained results show that topological state of any part of the trivial knot in a collapsed phase is almost trivial.
- Publication:
-
arXiv e-prints
- Pub Date:
- April 2002
- DOI:
- 10.48550/arXiv.cond-mat/0204149
- arXiv:
- arXiv:cond-mat/0204149
- Bibcode:
- 2002cond.mat..4149N
- Keywords:
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- Condensed Matter - Statistical Mechanics;
- Condensed Matter - Soft Condensed Matter;
- Mathematical Physics;
- Mathematics - Mathematical Physics
- E-Print:
- 15 pages, 6 eps-figures, LaTeX-RevTeX4