Bicolored tilings and the Scott map

TL;DR

This paper introduces bicolored tilings, defines transformations and an equivalence relation, and establishes a bijection with generalized Postnikov diagrams via the Scott map, linking geometric exchanges to edge flips.

Contribution

It presents a novel framework connecting bicolored tilings with Postnikov diagrams through the Scott map, expanding understanding of their combinatorial and geometric relationships.

Findings

Scott map creates a bijection between tilings and diagrams

Transformations preserve tiling equivalences

Edge flips correspond to geometric exchanges

Abstract

We define bicolored tilings as a disk with a collection of smooth curves with a coloring map on the tiles that these curves delimit. Using two transformations, we define an equivalence on tilings. We then define the Scott map which creates a bijection between a generalised version of Postnikov diagrams and bicolored tilings, preserving equivalences, and mapping the geometric exchange of a diagram to an edge flip in a tiling.