  Plane perfect matchings of $2n$ points in convex position are in bijection
with triangulations of convex polygons of size $n+2$. Edge flips are a classic
operation to perform local changes both structures have in common. In this
work, we use the explicit bijection from Aichholzer et al. (2018) to determine
the effect of an edge flip on the one side of the bijection to the other side,
that is, we show how the two different types of edge flips are related.
Moreover, we give an algebraic interpretation of the flip graph of
triangulations in terms of elements of the corresponding Temperley-Lieb
algebra.
