Gordian Unknots

Numerical simulations indicate that there exist conformations of the unknot, tied on a finite piece of rope, entangled in such a manner, that they cannot be disentangled to the torus conformation without cutting the rope. The simplest example of such a gordian unknot is presented. Knots are closed, self-avoiding curves in the 3-dimensional space. The shape and size of a knot, i.e. its conformation, can be changed in a very broad range without changing the knot type. The necessary condition to keep the knot type intact is that during all transformations applied to the knot the curve must remain self-avoiding. From the topological point of view, all conformations of a knot are equivalent but if the knot is considered as a physical object, it may be not so. Let us give a simple example. Take a concrete, knotted space curve K. Imagine, that K is inflated into a tube of diameter D. If K is scaled down without scaling down D, then there is obviously a minimum size below which one cannot go without changing the shape of K. Diminishing, in a thought or computer experiment, the size of a knot one arrives to the limit below which in some places of the knot the impenetrability of the tube on which it has been tied would be violated. Consider a knot tied on a piece of a rope. If the knot is tied in a loose manner, one can easily change its shape. However, the range of transformations available in such a process is much more narrow than in the case of knots tied on an infinitely thin rope. Limitations imposed on the transformations used to change the knot shape by the fixed thickness and length of the rope may make some conformations of the knot inaccessible from each other. The limitations can be in an elegant manner represented by the single condition that the global curvature of the knot cannot be larger than 2/D [1]. That it is the case we shall try to demonstrate in the most simple case of the unknot. The knot is a particular one since we know for it the shape of the ideal, least rope consuming conformation [2]. The simplest shape of the unknot is obviously circular. If the knot is tied on the rope of diameter D the shortest piece of rope one must use to 1