Combinatorial rigidity of 3-dimensional simplicial polytopes

TL;DR

This paper investigates the combinatorial rigidity of 3-dimensional simplicial polytopes, establishing necessary conditions for reducible cases and identifying some rigid examples, linking combinatorial structure to algebraic invariants.

Contribution

It introduces necessary conditions for combinatorial rigidity in reducible 3D simplicial polytopes and provides examples of rigid polytopes, advancing understanding of their algebraic-combinatorial properties.

Findings

Necessary condition for rigidity in reducible polytopes

Identification of some rigid reducible simplicial polytopes

Link between Betti numbers and combinatorial structure

Abstract

A simplicial polytope is combinatorially rigid if its combinatorial structure is determined by its graded Betti numbers which are important invariant coming from combinatorial commutative algebra. We find a necessary condition to be combinatorially rigid for 3-dimensional reducible simplicial polytopes and provide some rigid reducible simplicial polytopes.