# Example of C-rigid polytopes which are not B-rigid

## TL;DR

This paper investigates the relationship between B-rigidity and C-rigidity in simple polytopes, demonstrating that B-rigidity does not necessarily imply C-rigidity, thus clarifying their distinction.

## Contribution

It provides the first known example of C-rigid polytopes that are not B-rigid, showing these concepts are not equivalent.

## Key findings

- B-rigid simple polytopes are not always C-rigid
- C-rigidity is characterized by cohomology rings of quasitoric manifolds
- B-rigidity is characterized by Tor-algebra of the polytope

## Abstract

A simple polytope $P$ is said to be \emph{B-rigid} if its combinatorial structure is characterized by its Tor-algebra, and is said to be \emph{C-rigid} if its combinatorial structure is characterized by the cohomology ring of a quasitoric manifold over $P$. It is known that a B-rigid simple polytope is C-rigid. In this paper, we, further, show that the B-rigidity is not equivalent to the C-rigidity.