Scaling behavior of knotted random polygons and self-avoiding polygons: Topological swelling with enhanced exponent

Abstract

We show that the average size of self-avoiding polygons (SAP) with a fixed knot is much larger than that of no topological constraint if the excluded volume is small and the number of segments is large. We call it topological swelling. We argue an "enhancement" of the scaling exponent for random polygons with a fixed knot. We study them systematically through SAP consisting of hard cylindrical segments with various different values of the radius of segments. Here we mean by the average size the mean-square radius of gyration. Furthermore, we show numerically that the equilibrium length of a composite knot is given by the sum of those of all constituent prime knots. Here we define the equilibrium length of a knot by such a number of segments that topological entropic repulsions are balanced with the knot complexity in the average size. The additivity suggests the local knot picture.

arXiv comment33 pages, 7 figures. arXiv admin note: substantial text overlap with arXiv:1704.07510

References in corpus (17)

• Critical Exponents of the N-vector model
• Critical Exponents, Hyperscaling and Universal Amplitude Ratios for Two- and Three-Dimensional Self-Avoiding Walks
• Accurate estimate of the critical exponent $\nu$ for self-avoiding walks via a fast implementation of the pivot algorithm
• Critical exponents for random knots
• What is the length of a knot in a polymer?
• Topologically Driven Swelling of a Polymer Loop
• Equilibrium size of large ring molecules
• On the size of knots in ring polymers
• A New Algorithm for Numerical Calculation of Link Invariants
• Structure and dynamics of ring polymers: entanglement effects because of solution density and ring topology
• Average size of random polygons with fixed knot topology
• On the Limits of Analogy Between Self-Avoidance and Topology-Driven Swelling of Polymer Loops
• Finite-size and asymptotic behaviors of the gyration radius of knotted cylindrical polygons
• Characteristic length of random knotting for cylindrical self-avoiding polygons
• Gyration radius of a circular polymer under a topological constraint with excluded volume
• Knotting probability of self-avoiding polygons under a topological constraint
• Exponents of interchain correlation for self-avoiding walks and knotted self-avoiding polygons