Topological correlations in trivial knots: new arguments in support of the crumpled polymer globule

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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 Lᵥ × Lₕ of widths Lᵥ=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 Lₕ ≫ Lᵥ the knot complexity n is proportional to the length of the strip Lₕ. At the same time, the typical complexity of the quasi--knot which is a part of trivial knot behaves as n √Lₕ 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.