Cyclohedron

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

The cyclohedron is useful in studying 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 -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.

Properties

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.

gollark: Oh, that reminds me, NFTs are really stupidly implemented most of the time too.
gollark: * either → any
gollark: The "cryptocurrencies" without either of those are stupid and not decentralized.
gollark: Specifically: proof of stake is basically built-in compounding inequality; proof of space burns disks instead.
gollark: Proof of work is rather awful because it actively requires burning compute for no value, but all the alternatives are really bad too.

See also

References
  1. Bott, Raoul; Taubes, Clifford (1994). "On the self‐linking of knots". Journal of Mathematical Physics. 35 (10): 5247–5287. doi:10.1063/1.530750. MR 1295465.
  2. Markl, Martin (1999). "Simplex, associahedron, and cyclohedron". Contemporary Mathematics. 227: 235–265. doi:10.1090/conm/227. MR 1665469.
  3. Simion, Rodica (2003). "A type-B associahedron". Advances in Applied Mathematics. 30: 2–25. doi:10.1016/S0196-8858(02)00522-5.
  4. Stasheff, Jim (1997), "From operads to 'physically' inspired theories", in Loday, Jean-Louis; Stasheff, James D.; Voronov, Alexander A. (eds.), Operads: Proceedings of Renaissance Conferences, Contemporary Mathematics, 202, AMS Bookstore, pp. 53–82, ISBN 978-0-8218-0513-8, retrieved 1 May 2011
  5. Chapoton, Frédéric; Sergey, Fomin; Zelevinsky, Andrei (2002). "Polytopal realizations of generalized associahedra". Canadian Mathematical Bulletin. 45: 537–566. arXiv:math/0202004. doi:10.4153/CMB-2002-054-1.
  6. Carr, Michael; Devadoss, Satyan (2006). "Coxeter complexes and graph-associahedra". Topology and its Applications. 153: 2155–2168. doi:10.1016/j.topol.2005.08.010.

Further reading
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