Elementary moves on lattice polytopes

Abstract

We introduce a graph structure on Euclidean polytopes. The vertices of this graph are the $d$-dimensional polytopes contained in ${\mathbb{R}}^d$ and its edges connect any two polytopes that can be obtained from one another by either inserting or deleting a vertex, while keeping their vertex sets otherwise unaffected. We prove several results on the connectivity of this graph, and on a number of its subgraphs. We are especially interested in several families of subgraphs induced by lattice polytopes, such as the subgraphs induced by the lattice polytopes with $n$ or $n+1$ vertices, that turn out to exhibit intriguing properties.

35 pages, 9 figures

References in corpus (8)

• Enumeration of lattice 3-polytopes by their number of lattice points
• Improved bounds on the diameter of lattice polytopes
• Few smooth d-polytopes with n lattice points
• Classification of empty lattice $4$-simplices of width larger than two
• Flip-graph moduli spaces of filling surfaces
• Non-spanning lattice 3-polytopes
• On the width of lattice-free simplices
• On the diameter of lattice polytopes

Cited by in corpus (1)

• The pyramidal growth