Cyclohedron

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

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 cyclohedron appears in the study of 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 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

The 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 Δ {\displaystyle \Delta } of that graph is given by

lim d → ∞ Δ d = 5 2 {\displaystyle \lim _{d\rightarrow \infty }{\frac {\Delta }{d}}={\frac {5}{2}}} .[8]

See also

  • Associahedron
  • Permutohedron
  • Permutoassociahedron

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. ISBN 9780821809136. MR 1665469.
  3. ^ 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.
  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, 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
  5. ^ Chapoton, Frédéric; 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.
  6. ^ Carr, Michael; Devadoss, Satyan (2006). "Coxeter complexes and graph-associahedra". Topology and Its Applications. 153 (12): 2155–2168. arXiv:math/0407229. doi:10.1016/j.topol.2005.08.010.
  7. ^ Postnikov, Alexander (2009). "Permutohedra, Associahedra, and Beyond". International Mathematics Research Notices. 2009 (6): 1026–1106. arXiv:math/0507163. doi:10.1093/imrn/rnn153.
  8. ^ Pournin, Lionel (2017). "The asymptotic diameter of cyclohedra". Israel Journal of Mathematics. 219: 609–635. arXiv:1410.5259. doi:10.1007/s11856-017-1492-0.

Further reading

External links