NN-flips in even triangulations on the torus and Dehn twists preserving monodromies

In this paper, we show that any two even triangulations on the torus with the same and sufficiently large number of vertices can be transformed into each other by a sequence of two specifically defined deformations called an NN-flip and a P2P2-flip, up to isotopy, if and only if they have the same monodromy.