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

TL;DR

This paper generalizes Scott's map from polygon triangulations to surface tilings with marked points, showing that the fibers correspond to flip equivalence classes and exploring the map's combinatorial properties.

Contribution

It introduces a new association of strand diagrams to surface tilings, extending Scott's method, and proves the fibers of the Scott map are flip equivalence classes in polygon cases.

Findings

Fibers of the Scott map are flip equivalence classes in polygon tilings.

The size of the image relates to classical combinatorial problems.

Determined the size of the image in low ranks.

Abstract

We associate strand diagrams to tilings of surfaces with marked points, generalising Scott's method for triangulations of polygons. We thus obtain a map from tilings of surfaces to permutations of the marked points on boundary components, the {\em 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, and hence to determine the size in low ranks.