Cyclohedron

In geometry, the cyclohedron is a d-dimensional polytope where d can be any non-negative integer. It was first introduced as a combinatorial object by Raoul Bott and Clifford Taubes^[1] and, for this reason, it is also sometimes called the Bott–Taubes polytope. It was later constructed as a polytope by Martin Markl^[2] and by Rodica Simion.^[3] Rodica Simion describes this polytope as an associahedron of type B.

The 2-dimensional cyclohedron W₃ and the correspondence between its vertices and edges with a cycle on three vertices

The cyclohedron appears in the study of knot invariants.^[4]

Construction

Cyclohedra belong to several larger families of polytopes, each providing a general construction. For instance, the cyclohedron belongs to the generalized associahedra^[5] that arise from cluster algebra, and to the graph-associahedra,^[6] a family of polytopes each corresponding to a graph. In the latter family, the graph corresponding to the ${\displaystyle d}$-dimensional cyclohedron is a cycle on ${\displaystyle d+1}$ vertices.

In topological terms, the configuration space of ${\displaystyle d+1}$ distinct points on the circle ${\displaystyle S^{1}}$ is a ${\displaystyle (d+1)}$-dimensional manifold, which can be compactified into a manifold with corners by allowing the points to approach each other. This compactification can be factored as ${\displaystyle S^{1}\times W_{d+1}}$, where ${\displaystyle W_{d+1}}$ is the ${\displaystyle d}$-dimensional cyclohedron.

Just as the associahedron, the cyclohedron can be recovered by removing some of the facets of the permutohedron.^[7]

Properties

The 2-dimensional cyclohedron as the centrally symmetric triangulations of the regular hexagon

The graph made up of the vertices and edges of the ${\displaystyle d}$-dimensional cyclohedron is the flip graph of the centrally symmetric partial triangulations of a convex polygon with ${\displaystyle 2d+2}$ vertices.^[3] When ${\displaystyle d}$ goes to infinity, the asymptotic behavior of the diameter ${\displaystyle \Delta }$ of that graph is given by

${\displaystyle \lim _{d\rightarrow \infty }{\frac {\Delta }{d}}={\frac {5}{2}}}$.^[8]

See also

• Associahedron
• Permutohedron
• Permutoassociahedron