Cyclohedron

{{short description|Polytope associated with combinatorial problems}}

In geometry, the cyclohedron is a {{mvar|d}}-dimensional polytope where {{mvar|d}} can be any non-negative integer. It was first introduced as a combinatorial object by Raoul Bott and Clifford Taubes{{cite journal

| last1=Bott

| first1=Raoul

| author1-link=Raoul Bott

| last2=Taubes

| first2=Clifford

| author2-link=Clifford Taubes

| title=On the self‐linking of knots

| journal=Journal of Mathematical Physics

| volume=35

| issue=10

| year=1994

| doi=10.1063/1.530750

| pages=5247–5287

| mr=1295465}} and, for this reason, it is also sometimes called the Bott–Taubes polytope. It was later constructed as a polytope by Martin Markl{{cite journal

| last1=Markl

| first1=Martin

| title=Simplex, associahedron, and cyclohedron

| journal=Contemporary Mathematics

| volume=227

| year=1999

| doi=10.1090/conm/227

| pages=235–265

| isbn=9780821809136

| mr=1665469}} and by Rodica Simion.{{cite journal

| last1=Simion

| first1=Rodica

| author1-link=Rodica Simion

| title=A type-B associahedron

| journal=Advances in Applied Mathematics

| volume=30

| year=2003

| issue=1–2

| doi=10.1016/S0196-8858(02)00522-5 | doi-access=

| pages=2–25}} Rodica Simion describes this polytope as an associahedron of type B.

The cyclohedron appears in the study of knot invariants.{{Citation |last=Stasheff |first=Jim |authorlink=Jim Stasheff |year=1997 |chapter=From operads to 'physically' inspired theories |editor-last=Loday |editor-first=Jean-Louis |editor2-last=Stasheff |editor2-first=James D. |editor3-last=Voronov |editor3-first=Alexander A. |title=Operads: Proceedings of Renaissance Conferences |series=Contemporary Mathematics |volume=202 |pages=53–82 |publisher=AMS Bookstore |isbn=978-0-8218-0513-8 |chapter-url=http://www.math.unc.edu/Faculty/jds/operadchik.ps |accessdate=1 May 2011 |archive-date=23 May 1997 |archive-url=https://web.archive.org/web/19970523172846/http://www.math.unc.edu/Faculty/jds/operadchik.ps |url-status=dead }}

Construction

Cyclohedra belong to several larger families of polytopes, each providing a general construction. For instance, the cyclohedron belongs to the generalized associahedra{{cite journal

| last1=Chapoton

| first1=Frédéric

| last2=Sergey

| first2=Fomin

| author2-link=Sergey Fomin

| last3=Zelevinsky

| first3=Andrei

| author3-link=Andrei Zelevinsky

| title=Polytopal realizations of generalized associahedra

| journal=Canadian Mathematical Bulletin

| volume=45

| year=2002

| issue=4

| doi=10.4153/CMB-2002-054-1 | doi-access=free

| pages=537–566| arxiv=math/0202004

}} that arise from cluster algebra, and to the graph-associahedra,{{cite journal

| last1=Carr

| first1=Michael

| last2=Devadoss

| first2=Satyan

| author2-link=Satyan Devadoss

| title=Coxeter complexes and graph-associahedra

| journal=Topology and Its Applications

| volume=153

| year=2006

| issue=12

| doi=10.1016/j.topol.2005.08.010 | doi-access=free

| pages=2155–2168| arxiv=math/0407229

}} 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 $S^1 \times W_{d+1}$ , 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.{{cite journal

| first1=Alexander | last1=Postnikov

| title=Permutohedra, Associahedra, and Beyond

| journal=International Mathematics Research Notices

| volume=2009

| issue=6

| year=2009

| pages=1026–1106

| doi=10.1093/imrn/rnn153| arxiv=math/0507163

}}

Properties

The graph made up of the vertices and edges of the $d$ -dimensional cyclohedron is the flip graph of the centrally symmetric partial triangulations of a convex polygon with $2d+2$ vertices. When $d$ goes to infinity, the asymptotic behavior of the diameter $\Delta$ of that graph is given by

: $\lim_{d\rightarrow\infty}\frac{\Delta}{d}=\frac{5}{2}$ .{{cite journal

| first1=Lionel | last1=Pournin

| title=The asymptotic diameter of cyclohedra

| journal=Israel Journal of Mathematics

| volume=219

| year=2017

| pages=609–635

| doi=10.1007/s11856-017-1492-0 | doi-access=free| arxiv=1410.5259

}}

References

External links

{{mathworld|title=Cyclohedron|urlname=Cyclohedron|author=Bryan Jacobs}}

Category:Polytopes

Cyclohedron

Construction

Properties

See also

References

Further reading

External links