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Open Access Publications from the University of California

An Ergodic Algorithm for Generating Random Knots with a Thickness Constraint

  • Author(s): Chapman, Kyle Leland
  • Advisor(s): Millett, Kenneth
  • et al.

The first algorithm for sampling the space of thick equilateral knots, as a function of thickness, will be described. This algorithm is based on previous algorithms of applying random reflections.

To prove the usefulness of the algorithm, we create a method for turning any knot into the regular planar polygon using only thickness non-decreasing moves. This ensures that the algorithm has a positive probability of connecting any two knots with the required thickness constraint and so is ergodic. This ergodic sampling allows us to analyze the effects of thickness on properties of the geometric knot such as radius of gyration and knotting.

The data from this algorithm will show that the radius of gyration increases strongly with thickness, in that the growth exponent for radius of gyration increases with thickness. It also shows how knotting is decreased by the addition of a thickness constraint.

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