Scaling behavior of random knots

Abstract

Using numerical simulations we investigate how overall dimensions of random knots scale with their length. We demonstrate that when closed non-self-avoiding random trajectories are divided into groups consisting of individual knot types, then each such group shows the scaling exponent of Ϸ0.588 that is typical for self-avoiding walks. However, when all generated knots are grouped together, their scaling exponent becomes equal to 0.5 (as in non-self-avoiding random walks). We explain here this apparent paradox. We introduce the notion of the equilibrium length of individual types of knots and show its correlation with the length of ideal geometric representations of knots. We also demonstrate that overall dimensions of random knots with a given chain length follow the same order as dimensions of ideal geometric representations of knots.

The abundance of unknots in various models of polymer loops

“…5,30], are summarized in figures 1 and 4. The raw curves of probability given in figure 1 clearly show that the odds of finding an unknot in a set of loops get increasingly unfavorable as α decreases.…”

“…These works belong to the direction [26,27,28,29,30] addressing the spatial statistics of polymer loops restricted to remain in a certain topological knot state. It turns out that even for loops with no excluded volume and thus are not self-avoiding, N 0 marks the crossover scale between mostly Gaussian (N < N 0 ) and significantly non-Gaussian (N > N 0 ) statistics.…”

“…Indeed, at N < N 0 , locking the loop in the state of an unknot excludes only a small domain of the conformational space which produces only marginal (albeit non-trivial [25]) corrections to Gaussian statistics -for instance, mean-squared gyration radius of the loop is nearly linear in N . By contrast, at N > N 0 , the topological constraints are of paramount importance, making the loop statistics very much non-Gaussian, and consistent with effective self-avoidance [23,24,26,30].…”

Total Curvature and Total Torsion of Knotted Polymers

“…[1][2][3][4][5][6][7][8] Interestingly, the scaling profiles of such characteristics as the radius of gyration or the average crossing number of random polygons forming different knot types exhibit distinguishable profiles due to the different correcting terms that depend upon the topology. 3,6,7 This phenomenon is believed to reflect the physical behavior of cyclic polymers with different topologies under conditions where polymer segments that are not in direct contact neither attract nor repel each other. To understand better the statistical mechanics of polymers under these specific conditions, we use numerical simulations to determine the scaling profiles for the total curVature and total torsion of random closed polymers with fixed topology.…”

“…This type of random walk is frequently used to model polymers at thermodynamic equilibrium under θ-conditions or in melt phase where polymer segments that are not in a direct contact neither attract nor repel each other. [1][2][3][4][5][6][7][8][9] Linear chains without excluded volume are believed to behave as linear polymers in θ-conditions and have scaling exponent ν ) 0.5. The same scaling behavior is observed in the case of phantom polygons where the simulated segments can freely pass through each other.…”