# Stokes Polytopes

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## Description

Mathematicians studying combinatorics and topology have been obsessed with the associahedron for the past 30 years. During the SURE program of 2020 we researched a largely unknown set of shapes called Stokes Polytopes containing the associahedron. These shapes fall on a spectrum between the most simple shape, the n-dimensional cube, and the most complex, the n-dimensional associahedron. From papers written by Yuiley Barishnykov, Frederic Chapton, and many others we were able to understand the different ways these shapes are defined. These definitions include Q-compatibility and Hyperplanes derived from polyominoes. Q-compatibility takes two vertices of the shape given and asks if a line exists between them. This definition is great for exploring lower dimensional components (vertices and edges) of the shapes, but is not great if we want to know about the top dimensional (facets and ridges) components. The other definition takes a set of squares that are glued together such that a path of squares is formed, and uses a specific set of rules defined by Barishnykov to extract a set of equations. These equations when considered together are dual to the Stokes Polytope associated with the polyomino. This definition gives a more holistic view of the shape. Our goal is to find a good way to label these shapes and glue them together to create something that we call the Stokes Complex.

Stokes Polytopes

Mathematicians studying combinatorics and topology have been obsessed with the associahedron for the past 30 years. During the SURE program of 2020 we researched a largely unknown set of shapes called Stokes Polytopes containing the associahedron. These shapes fall on a spectrum between the most simple shape, the n-dimensional cube, and the most complex, the n-dimensional associahedron. From papers written by Yuiley Barishnykov, Frederic Chapton, and many others we were able to understand the different ways these shapes are defined. These definitions include Q-compatibility and Hyperplanes derived from polyominoes. Q-compatibility takes two vertices of the shape given and asks if a line exists between them. This definition is great for exploring lower dimensional components (vertices and edges) of the shapes, but is not great if we want to know about the top dimensional (facets and ridges) components. The other definition takes a set of squares that are glued together such that a path of squares is formed, and uses a specific set of rules defined by Barishnykov to extract a set of equations. These equations when considered together are dual to the Stokes Polytope associated with the polyomino. This definition gives a more holistic view of the shape. Our goal is to find a good way to label these shapes and glue them together to create something that we call the Stokes Complex.