"During the period from the end of the '60s through the beginning of the '70s, Conway pursued the objective of forming a complete table of knots. […] Therefore, he pulled another jewel from his bag of cornucopia and introduced the concept of tangle." - Murasugi
A $n$-tangle is an embedding of a collection of $n$ arcs in a 3-ball such that the endpoints are on specific 2$n$ points on the boundary sphere. The focus will be on 2-tangles. A rational tangle is a 2-tangle that is homeomorphic to the trivial tangle, which is formed by two unlinked arcs, vertical or horizontal. By taking the numerator closure of a tangle we obtain a knot and, in particular, the numerator of a rational tangle is a rational knot.
Rational tangles are associated, as the name suggests, to rational numbers (union infinity) and there is a deep connection between them and the theory of continued fractions. Apart from their mathematical importance in knot theory, rational tangles are crucial to the study of DNA topology and, in general, to biological applications of knot theory.