Cyclohedron

The [math]\displaystyle{ 2 }[/math]-dimensional cyclohedron [math]\displaystyle{ W_3 }[/math] and the correspondence between its vertices and edges with a cycle on three vertices

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

The cyclohedron appears in the study of knot invariants.

Construction

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

In topological terms, the configuration space of [math]\displaystyle{ d+1 }[/math] distinct points on the circle [math]\displaystyle{ S^1 }[/math] is a [math]\displaystyle{ (d+1) }[/math]-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 [math]\displaystyle{ S^1 \times W_{d+1} }[/math], where [math]\displaystyle{ W_{d+1} }[/math] is the [math]\displaystyle{ d }[/math]-dimensional cyclohedron.

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

Properties

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

[math]\displaystyle{ \lim_{d\rightarrow\infty}\frac{\Delta}{d}=\frac{5}{2} }[/math].

See also

• Associahedron
• Permutohedron
• Permutoassociahedron

Further reading

• Forcey, Stefan; Springfield, Derriell (December 2010), "Geometric combinatorial algebras: cyclohedron and simplex", Journal of Algebraic Combinatorics 32 (4): 597–627, doi:10.1007/s10801-010-0229-5 
• Morton, James; Pachter, Lior; Shiu, Anne; Sturmfels, Bernd (January 2007), "The Cyclohedron Test for Finding Periodic Genes in Time Course Expression Studies", Statistical Applications in Genetics and Molecular Biology 6 (1): Article 21, doi:10.2202/1544-6115.1286, PMID 17764440 

External links

• Bryan Jacobs. "Cyclohedron". http://mathworld.wolfram.com/Cyclohedron.html.