Inferring a Cell Structure on the Space of Cyclooctane Conformations

TL;DR

The article presents a symmetry-driven cell structure for the cyclooctane conformation space, revealing a sphere–Klein bottle decomposition in the labeled space and a contractible quotient under dihedral symmetries. By partitioning configurations according to symmetry types and analyzing them with Isomap and persistent homology, the authors derive a refined cellular model that matches observed topological invariants. Their work provides a concrete geometric-understanding framework for cyclooctane conformations and lays groundwork for studying conformational energies within the established cell complex. The results have potential implications for how molecular flexibility in cyclic alkanes is characterized and navigated computationally.

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.

Figures (20)

• Figure 1: A typical configucation of the cyclooctane linkage
• Figure 2: The Euclidean distance $d_\Vert$ and the angular distance $d_\angle$ are equivalent. The picture has been obtained by plotting 100k randomly choosen pairwise distances (out of the $\approx 18M$ possible ones).
• Figure 3: Persistent homology of $\mathit{Cfg}(K)$ with coefficients in $\mathbf{F}_2$ and $\mathbf{F}_3$ w.r.t. $d_\angle$ and $d_\Vert$. Each point $(b, d)$ corresponds to one homology interval $[b, d)$. For this sample, the diagrams for $H_\bullet(-, \mathbf{F}_2)$ and $H_\bullet(-, \mathbf{F}_3)$ coincide; this is not the case for all subspaces we consider.
• Figure 4: Isomap projection (top: w.r.t. $d_\Vert$, bottom: $d_\angle$) of $\mathit{Cfg}(K)$ and its subspaces $A$, $B$, $C$.
• Figure 5: Cellular model of the labeled configuration space $\mathit{Cfg}(K)$, and its subspaces $A$, $B$ and $C$. The subspace $A$ consists of two disjoint disks, $B$ is a cylinder, and $C$ is a model for a Klein bottle sewed together by two Möbius strips (both with the dashed line as boundary; one with core curve $a$, one with core curve $b$; see (\ref{['fig:model:C:2']}) for a different drawing of the same model). The three subspaces are glued together along the 1-cells $a$ and $b$. Thus, the whole space is the union of the sphere $S^2 \cong A \cup B$ the Klein bottle $C$, where $A \cup B$ and $C$ intersect in the two disjoint circles $a$ and $b$.

Theorems & Definitions (3)

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