Abstract
A method for generating families of particular 3D curves which differs from random self-avoiding walks is presented. This method is based on a chain code called knot numbers. The knot-number notation describes discrete knots. A discrete knot is the digitalized representation of a knot and is composed of constant orthogonal straight-line segments. The orthogonal direction changes of the contiguous straight-line segments of the knot define the chain elements. There are only five possible orthogonal direction changes for representing any discrete knot. Thus, the chain elements are considered as one base-five integer number (knot number). In this manner, we obtain a unique knot descriptor. This description is invariant under translation, rotation, starting point, and coding direction of the discrete knot.By evaluating all possible combinations of chain elements of curves and considering some restrictions, we obtain interesting families of curves which allow us to find the smallest nontrivial knot. Finally, we present a modest attempt at building the table of minimal discrete knots.
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