Biarcs, Global Radius of Curvature, and the Computation of Ideal Knot Shapes

M. Carlen, B. Laurie, J.H. Maddocks, J. Smutny

Chapter 5 (2005) 75-108 in "Physical and Numerical Models in Knot Theory and Their Application to the Life Sciences", Eds. J. Calvo, K. Millett, E. Rawdon, and A. Stasiak, published by World Scientific.

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Abstract
We combine the global radius of curvature characterisation of knot thickness, the biarc discretisation of space curves, and simulated annealing code to compute approximations to the ideal shapes of the trefoil and figure-eight knots. The computations contain no discretisation error, and give rigorous lower bounds on thickness of the true ideal shapes. The introduction of a precise definition of a contact set for an approximately ideal shape allows us to resolve previously unobserved features. For example, in our approximations of both the ideal trefoil and figure-eight knots, local curvature is within a rather small tolerance of being active, i.e. achieving thickness, at several points along the knot.