Biarcs, Global Radius of Curvature, and the Computation of Ideal
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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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.