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
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.
Just as the associahedron, the cyclohedron can be recovered by removing some of the facets of the permutohedron.[7]
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.[3] When goes to infinity, the asymptotic behavior of the diameter of that graph is given by
^Stasheff, Jim (1997), , in Loday, Jean-Louis; Stasheff, James D.; Voronov, Alexander A. (eds.), Operads: Proceedings of Renaissance Conferences, Contemporary Mathematics, vol. 202, AMS Bookstore, pp. 53–82, ISBN978-0-8218-0513-8, archived from the original on 23 May 1997, retrieved 1 May 2011
cyclohedron, geometry, cyclohedron, displaystyle, dimensional, polytope, where, displaystyle, negative, integer, first, introduced, combinatorial, object, raoul, bott, clifford, taubes, this, reason, also, sometimes, called, bott, taubes, polytope, later, cons. In geometry the cyclohedron is a d displaystyle d dimensional polytope where d displaystyle 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 2 displaystyle 2 dimensional cyclohedron W 3 displaystyle W 3 and the correspondence between its vertices and edges with a cycle on three vertices The cyclohedron is useful in studying knot invariants 4 Contents 1 Construction 2 Properties 3 See also 4 References 5 Further reading 6 External linksConstruction EditCyclohedra 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 d displaystyle d dimensional cyclohedron is a cycle on d 1 displaystyle d 1 vertices In topological terms the configuration space of d 1 displaystyle d 1 distinct points on the circle S 1 displaystyle S 1 is a d 1 displaystyle 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 W d 1 displaystyle S 1 times W d 1 where W d 1 displaystyle W d 1 is the d displaystyle d dimensional cyclohedron Just as the associahedron the cyclohedron can be recovered by removing some of the facets of the permutohedron 7 Properties EditThe graph made up of the vertices and edges of the d displaystyle d dimensional cyclohedron is the flip graph of the centrally symmetric triangulations of a convex polygon with 2 d 2 displaystyle 2d 2 vertices 3 When d displaystyle d goes to infinity the asymptotic behavior of the diameter D displaystyle Delta of that graph is given by lim d D d 5 2 displaystyle lim d rightarrow infty frac Delta d frac 5 2 8 See also EditAssociahedron Permutohedron PermutoassociahedronReferences Edit 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 Markl Martin 1999 Simplex associahedron and cyclohedron Contemporary Mathematics 227 235 265 doi 10 1090 conm 227 ISBN 9780821809136 MR 1665469 a b Simion Rodica 2003 A type B associahedron Advances in Applied Mathematics 30 1 2 2 25 doi 10 1016 S0196 8858 02 00522 5 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 vol 202 AMS Bookstore pp 53 82 ISBN 978 0 8218 0513 8 archived from the original on 23 May 1997 retrieved 1 May 2011 Chapoton Frederic Sergey Fomin Zelevinsky Andrei 2002 Polytopal realizations of generalized associahedra Canadian Mathematical Bulletin 45 4 537 566 arXiv math 0202004 doi 10 4153 CMB 2002 054 1 Carr Michael Devadoss Satyan 2006 Coxeter complexes and graph associahedra Topology and Its Applications 153 12 2155 2168 doi 10 1016 j topol 2005 08 010 Postnikov Alexander 2009 Permutohedra Associahedra and Beyond International Mathematics Research Notices 2009 6 1026 1106 arXiv math 0507163 doi 10 1093 imrn rnn153 Pournin Lionel 2017 The asymptotic diameter of cyclohedra Israel Journal of Mathematics 219 609 635 doi 10 1007 s11856 017 1492 0 Further reading EditForcey Stefan Springfield Derriell December 2010 Geometric combinatorial algebras cyclohedron and simplex Journal of Algebraic Combinatorics 32 4 597 627 arXiv 0908 3111 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 arXiv q bio 0702049 doi 10 2202 1544 6115 1286 PMID 17764440External links EditBryan Jacobs Cyclohedron MathWorld Retrieved from https en wikipedia org w index php title Cyclohedron amp oldid 1113926599, wikipedia, wiki, book, books, library,
