Rational tangles and the modular group

TL;DR

This paper explores the relationship between rational tangles, the modular group, and the braid group, establishing a correspondence that links isotopy classes of tangles to rational numbers.

Contribution

It introduces a natural association between transformations of rational tangles and elements of the modular group, clarifying their algebraic and topological connections.

Findings

Isotopy classes of rational tangles correspond to rational numbers.

Transformations of tangles relate to elements of the modular group.

Connection between rational tangles, the modular group, and the braid group B3.

Abstract

There is a natural way to associate with a transformation of an isotopy class of rational tangles to another, an element of the modular group. The correspondence between the isotopy classes of rational tangles and rational numbers follows, as well as the relation with the braid group $B_3$.