Rational Tangles and the Modular Group

Abstract

There is a natural way to define an isomorphism between the group of transformations of isotopy classes of rational tangles and the modular group. This isomorphism allows to give a simple proof of the Conway theorem, stating the one-to-one correspondence between isotopy classes of rational tangles and rational numbers. Two other simple ways to define this isomorphisms, one of which suggested by Arnold, are also shown. Introduction A tangle with four ends (here called simply tangle) is an embedding of two closed segments in a ball, such that their endpoints are four distinct points of the bounding sphere, and the image of the interior of the segments lie at the interior of the ball. Figure 1. Two tangles Two tangles are said isotopic if one can be deformed continuously to the other in the set of tangles with fixed endpoints. A tangle is said rational if can be deformed continuously, in the set of tangles with non fixed endpoints, to a tangle consisting of two unlinked and unknotted segments. Examples. The tangles in Figure 1 are not rational, since the white strand of the left tangle is knotted, and the two strands of the tangle at right cannot be unlinked if we allow the endpoints of the strands to move on the bounding sphere. See Figure 2 for an example of rational tangle. The main result on rational tangles is a theorem by Conway (announced in [2]) stating that it is possible to associate with every rational tangle one and only one rational number, so that two rational tangles are isotopy equivalent if and only if they are represented by the same rational number. The idea of the elementary and self-consistent proof of this theorem given in [3] is that any rational tangle is isotopy equivalent to exactly one canonical rational tangle (alternating). To such a tangle one associates exactly one continued fraction, and hence a rational number; conversely, from a rational number, via its continued fraction, the canonical representative of exactly one class of rational tangles is constructed.