In geometry, the cyclohedron is a -dimensional polytope where 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 cyclohedron appears in the study of knot invariants. [4]
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 -dimensional cyclohedron is a cycle on vertices.
In topological terms, the configuration space of distinct points on the circle is a -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 , where is the -dimensional cyclohedron.
Just as the associahedron, the cyclohedron can be recovered by removing some of the facets of the permutohedron. [7]
The graph made up of the vertices and edges of the -dimensional cyclohedron is the flip graph of the centrally symmetric triangulations of a convex polygon with vertices. [3] When goes to infinity, the asymptotic behavior of the diameter of that graph is given by
In geometry, the cyclohedron is a -dimensional polytope where 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 cyclohedron appears in the study of knot invariants. [4]
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 -dimensional cyclohedron is a cycle on vertices.
In topological terms, the configuration space of distinct points on the circle is a -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 , where is the -dimensional cyclohedron.
Just as the associahedron, the cyclohedron can be recovered by removing some of the facets of the permutohedron. [7]
The graph made up of the vertices and edges of the -dimensional cyclohedron is the flip graph of the centrally symmetric triangulations of a convex polygon with vertices. [3] When goes to infinity, the asymptotic behavior of the diameter of that graph is given by