Knotting probability of self-avoiding polygons under a topological constraint

Area of Science:

• Polymer physics
• Statistical mechanics
• Knot theory

Background:

• Knotting probability quantifies the likelihood of a random polymer forming a specific topological knot.
• Self-avoiding polygons (SAPs) are standard models for polymers, but incorporating excluded volume effects is crucial for realistic simulations.

Purpose of the Study:

• To investigate the knotting probability of knots in cylindrical self-avoiding polygons (SAPs) as a function of segment number and excluded volume.
• To establish a compact formula describing knotting probabilities and explore their dependence on excluded volume.

Main Methods:

• Numerical simulations of cylindrical self-avoiding polygons (SAPs) with hard cylindrical segments of unit length and radius r_ex.
• Analysis of knotting probabilities for various prime and composite knots.
• Derivation of a sum rule based on the factorization property of knotting probabilities.

Main Results:

• A compact formula accurately describes knotting probabilities for cylindrical SAPs across different segment numbers (N) and excluded volumes (r_ex).
• Increasing excluded volume generally decreases knotting probability for prime knots, except for the trefoil knot.
• The trefoil knot and its related knots become dominant in larger excluded volumes, suggesting a local knotting picture.

Conclusions:

• The study provides a unified description of knotting probabilities in polymers with excluded volume, connecting small and large segment behaviors.
• The findings suggest that excluded volume plays a critical role in determining knot topology in polymers.
• The cylindrical SAP model offers insights into the knotting of circular DNA, with r_ex analogous to the screening length.