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Equivalence Between Four Models of Associahedra

Somnath Basu, Sandip Samanta

TL;DR

The paper addresses the problem of unifying multiple realizations of the associahedron by proving explicit combinatorial isomorphisms among four models: Stasheff complexes ($K_n$), Loday's cone construction, collapsed multiplihedra ($J'_n$), and graph cubeahedra for path graphs ($\mathcal{C}P_n$). It introduces an equivalent formulation of multiplihedra and establishes three sequential isomorphisms (Loday ↔ Stasheff, Stasheff ↔ collapsed multiplihedra, collapsed multiplihedra ↔ graph cubeahedra) to conclude that all four models are combinatorially identical via face posets. The approach relies on constructing explicit bijections between faces of corresponding codimensions and by translating between bracketings, painted trees, and design tubings. This unifies diverse realizations of the same combinatorial type and enhances connections to $A_\infty$ structures, moduli spaces, and graph associahedra. The work provides a cohesive framework that links classical Tamari/Stasheff pictures with Loday's convex realizations and modern graph-theoretic polytopes, enabling broader applications in discrete geometry and homotopy theory.

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.

Equivalence Between Four Models of Associahedra

TL;DR

The paper addresses the problem of unifying multiple realizations of the associahedron by proving explicit combinatorial isomorphisms among four models: Stasheff complexes (), Loday's cone construction, collapsed multiplihedra (), and graph cubeahedra for path graphs (). It introduces an equivalent formulation of multiplihedra and establishes three sequential isomorphisms (Loday ↔ Stasheff, Stasheff ↔ collapsed multiplihedra, collapsed multiplihedra ↔ graph cubeahedra) to conclude that all four models are combinatorially identical via face posets. The approach relies on constructing explicit bijections between faces of corresponding codimensions and by translating between bracketings, painted trees, and design tubings. This unifies diverse realizations of the same combinatorial type and enhances connections to structures, moduli spaces, and graph associahedra. The work provides a cohesive framework that links classical Tamari/Stasheff pictures with Loday's convex realizations and modern graph-theoretic polytopes, enabling broader applications in discrete geometry and homotopy theory.

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 maps. We also prove a combinatorial isomorphism between Stasheff associahedra, collapsed multiplihedra and graph cubeahedra for path graphs.
Paper Structure (10 sections, 11 theorems, 22 equations, 13 figures)

This paper contains 10 sections, 11 theorems, 22 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Description of Four Models of Associahedra
  3. Stasheff complexes
  4. Loday's Cone Construction
  5. Collapsed Multiplihedra
  6. Graph Cubeahedra and Design Tubings
  7. Isomorphisms Between The Four Models
  8. Loday's construction vs Stasheff complexes
  9. Stasheff complexes vs Collapsed Multiplihedra
  10. Collapsed Multiplihedra vs Graph Cubeahedra

Key Result

Proposition 2.1

HAH$K_{i}$ is homeomorphic to $I^{i-2}$ and degeneracy maps $s_{j}: K_{i} \rightarrow K_{i-1}$ for $1 \leq j \leq i$ can be defined so that the following relations hold:

Figures (13)

  • Figure 1: Earliest realizations of associahedra
  • Figure 2: Correspondence between bracketing and rooted binary tree
  • Figure 3: Loday's embedded $K_5$ in $\mathbb{R}^3$
  • Figure 7: $\mathcal{J}(4)$ and its degeneration to $K_5$
  • Figure 8: Admissible nodes
  • ...and 8 more figures

Theorems & Definitions (31)

  • Proposition 2.1
  • Definition 2.2: $A_n$ form and $A_n$ space
  • Definition 2.3: Associahedron
  • Lemma 2.4
  • Theorem 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 21 more