Inferring a Cell Structure on the Space of Cyclooctane Conformations

TL;DR

This paper introduces a natural cell structure for the conformation space of cyclooctane, revealing its topological properties and contrasting labeled and unlabeled variants using symmetry-based partitioning and topological analysis.

Contribution

It proposes a novel cell decomposition of cyclooctane's conformation space based on symmetry, and analyzes its topology, showing the unlabeled space is contractible.

Findings

Unlabeled conformation space is contractible.

Cell structure based on symmetry patterns.

Method combines dimensionality reduction and topological data analysis.

Abstract

The conformation space of cyclooctane, a ringlike organic molecule comprising eight carbon atoms, is a two-dimensional algebraic variety, which has been studied extensively for more than 90 years. We propose a cell structure representing this space, which arises naturally by partitioning the space into subsets of conformations that admit particular symmetries. We do so both for the labeled conformation space, in which the carbon atoms are considered as distinct, and for the actual, unlabeled, conformation space. The proposed cell structure is obtained by identifying subspaces of conformations based on symmetry patterns and studying the geometry and topology of these subsets using methods from dimensionality reduction and topological data analysis. Our findings suggest that, in contrast to the labeled variant, the conformation space of cyclooctane is contractible.