The fibres of the Scott map on polygon tilings are the flip equivalence classes

We define a map from tilings of surfaces with marked points to strand diagrams, generalising Scott’s construction for the case of triangulations of polygons. We thus obtain a map from tilings of surfaces to permutations of the marked points on boundary components, the Scott map. In the disk case (polygon tilings) we prove that the fibres of the Scott map are the flip equivalence classes. The result allows us to consider the size of the image as a generalisation of a classical combinatorial problem. We hence determine the size in low ranks.