  We show that under certain conditions the flyping operation on rational
tangles, which produces topologically isotopic tangles, may also produce
tangles which are not Legendrian isotopic when viewed in the standard contact
structure on $\mathbb{R}^3$. This work is motivated by questions posed by
Traynor, and incorporates techniques inspired by work of Eliashberg and Fraser.
The proofs within employ a new method of diagramming rational tangles which is
also used to easily describe the characteristic foliations of compressing discs
for their complements.
