  This paper gives infinitely many examples of unknot diagrams that are hard,
in the sense that the diagrams need to be made more complicated by Reidemeister
moves before they can be simplified. In order to construct these diagrams, we
prove theorems characterizing when the numerator of the sum of two rational
tangles is an unknot. The key theorem shows that the numerator of the sum of
two rational tangles [P/Q] and [R/S] is unknotted if and only if PS + QR has
absolute value equal to 1. The paper uses these results in studying processive
DNA recombination, finding minimal size unknot diagrams, generalizing to
collapses to knots as well as to unknots, and in finding unknots with
arbirarily high complexity. The paper is self-contained, with a review of the
theory of rational tangles and a last section on relationships of the theme of
the paper with other aspects of topology and number theory. The present version
of the paper contains a correction to Figure 22 of the previous (and published)
version, a small expansion to the section on Farey fractions, and an updated
reference to the paper "Unknotting unknots" by Henrich and Kauffman.
