Cyclohedron
In geometry, the cyclohedron is a formula_1-dimensional polytope where formula_1 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 is useful in studying 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 formula_1-dimensional cyclohedron is a cycle on formula_4 vertices. In topological terms, the configuration space of formula_4 distinct points on the circle formula_6 is a formula_7-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 formula_8, where formula_9 is the formula_1-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 formula_1-dimensional cyclohedron is the flip graph of the centrally symmetric triangulations of a convex polygon with formula_12 vertices. When formula_1 goes to infinity, the asymptotic behavior of the diameter formula_14 of that graph is given by