Cyclohedron Explained

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 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^[4] that arise from cluster algebra, and to the graph-associahedra,^[5] a family of polytopes each corresponding to a graph. In the latter family, the graph corresponding to the

d -dimensional cyclohedron is a cycle on d +1 vertices.

In topological terms, the configuration space of

d +1 distinct points on the circle S ^{1} is a (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 S ^{1} x W _{d+1} , where W _{d+1} is the d -dimensional cyclohedron.

Just as the associahedron, the cyclohedron can be recovered by removing some of the facets of the permutohedron.^[6]

Properties

The graph made up of the vertices and edges of the

d -dimensional cyclohedron is the flip graph of the centrally symmetric triangulations of a convex polygon with 2 d +2 vertices.^[3] When d goes to infinity, the asymptotic behavior of the diameter \Delta of that graph is given by\lim _{d → infty} (\Delta)/(= d) (5)/(2) .^[7]

See also

• Associahedron
• Permutohedron
• Permutoassociahedron

Notes and References

1. Bott. Raoul. Raoul Bott. Taubes. Clifford. Clifford Taubes. On the self‐linking of knots. Journal of Mathematical Physics. 35. 10. 1994. 10.1063/1.530750. 5247–5287. 1295465.
2. Markl. Martin. Simplex, associahedron, and cyclohedron. Contemporary Mathematics. 227. 1999. 10.1090/conm/227. 235–265. 9780821809136. 1665469.
3. Simion. Rodica. Rodica Simion. A type-B associahedron. Advances in Applied Mathematics. 30. 2003. 1–2. 10.1016/S0196-8858(02)00522-5 . 2–25.
4. Chapoton. Frédéric. Sergey. Fomin. Sergey Fomin. Zelevinsky. Andrei. Andrei Zelevinsky. Polytopal realizations of generalized associahedra. Canadian Mathematical Bulletin. 45. 2002. 4. 10.4153/CMB-2002-054-1 . free. 537–566. math/0202004.
5. Carr. Michael. Devadoss. Satyan. Satyan Devadoss. Coxeter complexes and graph-associahedra. Topology and Its Applications. 153. 2006. 12. 10.1016/j.topol.2005.08.010 . free. 2155–2168. math/0407229.
6. Alexander . Postnikov . Permutohedra, Associahedra, and Beyond. International Mathematics Research Notices. 2009. 6. 2009. 1026–1106. 10.1093/imrn/rnn153. math/0507163.
7. Lionel . Pournin. The asymptotic diameter of cyclohedra. Israel Journal of Mathematics. 219. 2017. 609–635. 10.1007/s11856-017-1492-0 . free. 1410.5259.