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Discrete Curvatures and Convex Polytopes

Jesús A. De Loera, Jillian Eddy, Sawyer Jack Robertson, José Alejandro Samper

Abstract

We study Forman--Ricci and effective resistance curvatures on the skeleta of convex polytopes. Our guiding questions are: how frequently do polytopal graphs exhibit everywhere positive curvature, and what structural constraints does positivity impose? For Forman--Ricci curvature we derive an exact identity for the average edge curvature in terms of flag $f$-numbers and establish the existence of infinite families of Forman--Ricci-positive polytopes in every fixed dimension $d\ge 6$. We prove finiteness results in low dimension: there are only finitely many Forman--Ricci-positive $3$- and $4$-polytopes; for $d=5$ we show finiteness in the simplicial case, and conjecture its extension to $5$-polytopes more generally. For the resistance curvature $κ(v)$ we establish the existence of infinite families for all $d\ge 3$, and we provide a quantitative lower bound for $κ(v)$ in a simple $3$-polytope in terms of the lengths of the three $2$-faces incident to $v$. This bound leads to constructions of non-vertex-transitive, resistance-positive $3$-polytopes via $Δ$-operations, and a degree-based obstruction showing that if each neighbor of $v$ has degree at most $d_v-2$, then $κ(v)\le 0$. Our results suggest that positive curvature on polytopal skeletons is rare and constrained.

Discrete Curvatures and Convex Polytopes

Abstract

We study Forman--Ricci and effective resistance curvatures on the skeleta of convex polytopes. Our guiding questions are: how frequently do polytopal graphs exhibit everywhere positive curvature, and what structural constraints does positivity impose? For Forman--Ricci curvature we derive an exact identity for the average edge curvature in terms of flag -numbers and establish the existence of infinite families of Forman--Ricci-positive polytopes in every fixed dimension . We prove finiteness results in low dimension: there are only finitely many Forman--Ricci-positive - and -polytopes; for we show finiteness in the simplicial case, and conjecture its extension to -polytopes more generally. For the resistance curvature we establish the existence of infinite families for all , and we provide a quantitative lower bound for in a simple -polytope in terms of the lengths of the three -faces incident to . This bound leads to constructions of non-vertex-transitive, resistance-positive -polytopes via -operations, and a degree-based obstruction showing that if each neighbor of has degree at most , then . Our results suggest that positive curvature on polytopal skeletons is rare and constrained.

Paper Structure

This paper contains 15 sections, 40 theorems, 135 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Notation and mathematical background
  4. Our Contributions
  5. Forman--Ricci curvature of polytopes
  6. Average curvature, symmetry, and high dimensions
  7. Forman--Ricci-positive polytopes and the average degree of a vertex
  8. Forman--Ricci curvature in Dimension 3
  9. Forman--Ricci curvature in Dimension $4$
  10. Curvature of simplicial polytopes and 5-dimensional polytopes
  11. Resistance Curvature
  12. Resistance bounds via path lengths
  13. Resistance-positive polytopes and $\Delta$-expansions
  14. Conditions for negative resistance curvature
  15. Computational Details for Example 3.7

Key Result

Theorem 1.3

Let $P$ be a polytope. Suppose there exists $c > 0$ such that for $\kappa_F\left({e}\right) \geq c$ for each edge $e\in E(P)$. Then the following hold:

Figures (5)

  • Figure 1: (a) Parallel neighbors (blue) of an edge $e$ (red) in the case where $e$ is an edge of a heptagon (left), and in the case where $e$ is adjacent to a vertex of degree 7 (right). (b) A $3$-dimensional square cupola polytope with edges labeled according to their Forman--Ricci curvature.
  • Figure 2: (a) The prism $3$-polytopes and their graphs with faces consisting of (in clockwise order) five, six, seven, and eight vertices. (b) The $3$-polytope constructed via its graph $G_k$ in \ref{['subsec:resistance-positive-polyhedra']}, here shown for $k=3, 5$.
  • Figure 3: Illustrations of the Schlegel diagrams used in the proof of \ref{['thm:3polytope-degree']}.
  • Figure 4: An illustration of the graphs of 49 Forman--Ricci-positive $3$-polytopes, obtained as a random sample of the Forman--Ricci-positive $3$-polytopes on at most twelve vertices.
  • Figure 5: (a) The $3$-simplex, vertex transitive and resistance positive.(b)--(d)$\Delta$-expansions of (a), resistance-positive via \ref{['thm:delta-expansions']}. (e) The $3$-cube, likewise vertex-transitive and resistance positive. (f) A $\Delta$-expansion of the $3$-cube that is resistance positive via \ref{['thm:delta-expansions']}. (g) A $\Delta$-expansion of the $3$-cube not covered by the sufficient criteria of \ref{['thm:delta-expansions']}, yet containing no forbidden face sequences (cf. \ref{['eq:forbidden']}). (h) A $\Delta$-expansion of the $3$-cube failing the same criteria and containing forbidden face sequences (cf. \ref{['eq:forbidden']}) which is resistance positive.

Theorems & Definitions (74)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: Bonnet-Myers Theorem for Forman--Ricci curvature (see forman)
  • Lemma 1.4
  • Definition 1.5
  • Theorem 1.6: Bonnet-Myers Theorem for Resistance Curvature (see DOS)
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 64 more