Plane perfect matchings of 2n points in convex position are known to be in bijection with triangulations of convex polygons of size n + 2; they are both counted by the Catalan numbers. We explain how to give a direct bijection and how it can be extended to a bijection between monochromatic matchings on k colours and tilings by (k+2)-gons. Edge flips are a classic operation to perform local changes in both sets. We use the above bijection to determine the two types of edge flips are related. We use this to give an algebraic interpretation of the flip graph of triangulations in terms of elements of the corresponding Temperley-Lieb algebra. This is joint work with O. Aichholzer, L. Donner (Andritsch), B. Vogtenhuber.