Equivalence Between Four Models of Associahedra

TL;DR

This paper establishes combinatorial isomorphisms between various models of associahedra, including Stasheff associahedra, multiplihedra, and graph cubeahedra, providing new insights into their structural relationships.

Contribution

It introduces a combinatorial isomorphism between Stasheff associahedra and Loday's cone construction, and describes multiplihedra and their relations to other polytopes.

Findings

Established isomorphism between Stasheff associahedra and Loday's cone construction

Provided an alternate description of multiplihedra

Proved isomorphisms among associahedra, multiplihedra, and graph cubeahedra

Abstract

We present a combinatorial isomorphism between Stasheff associahedra and an inductive cone construction of those complexes given by Loday. We give an alternate description of certain polytopes, known as multiplihedra, which arise in the study of $A_\infty$ maps. We also prove a combinatorial isomorphism between Stasheff associahedra, collapsed multiplihedra and graph cubeahedra for path graphs.