Marked tubes and the graph multiplihedron

Devadoss, Satyan L.; Forcey, Stefan

doi:10.2140/agt.2008.8.2081

Cited by 22 publications

(45 citation statements)

References 19 publications

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“…Parts (a)-(c) of Fig. 3 from [6] show examples of allowable tubings, whereas (d)-(f) depict the forbidden ones. Graph associahedra of a path and a disconnected graph.…”

Section: Review Of Some Geometric Combinatoricsmentioning

confidence: 99%

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Geometric combinatorial algebras: cyclohedron and simplex

Forcey¹,

Springfield²

2010

J Algebr Comb

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In this paper we report on results of our investigation into the algebraic structure supported by the combinatorial geometry of the cyclohedron. Our new graded algebra structures lie between two well known Hopf algebras: the MalvenutoReutenauer algebra of permutations and the Loday-Ronco algebra of binary trees. Connecting algebra maps arise from a new generalization of the Tonks projection from the permutohedron to the associahedron, which we discover via the viewpoint of the graph associahedra of Carr and Devadoss. At the same time, that viewpoint allows exciting geometrical insights into the multiplicative structure of the algebras involved. Extending the Tonks projection also reveals a new graded algebra structure on the simplices. Finally this latter is extended to a new graded Hopf algebra with basis all the faces of the simplices.

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“…Parts (a)-(c) of Fig. 3 from [6] show examples of allowable tubings, whereas (d)-(f) depict the forbidden ones. Graph associahedra of a path and a disconnected graph.…”

Section: Review Of Some Geometric Combinatoricsmentioning

confidence: 99%

“…There are many open questions regarding formulas for the face vectors of graph associahedra for specific types of graphs. Figure 4, partly from [6], shows two examples of graph associahedra. These have underlying graphs a path and a disconnected graph, respectively, with three nodes each.…”

Section: Review Of Some Geometric Combinatoricsmentioning

confidence: 99%

Geometric combinatorial algebras: cyclohedron and simplex

Forcey¹,

Springfield²

2010

J Algebr Comb

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“…Thus the binary painted trees are refinements of the trees having nodes of type (4)- (6). Minimal refinement refers to the covering relation in this poset: t minimally refines t means that t refines t and also that there is no t such that both t refines t and t refines t .…”

Section: Definitionmentioning

confidence: 99%

“…Another effort underway is the generalization of the multiplihedron and its quotients by analogy to the graph-associahedra introduced by Carr and Devadoss, in [5]. The graph-multiplihedra will be presented in [6].…”

Section: Introductionmentioning

confidence: 99%

Convex hull realizations of the multiplihedra

Forcey

2008

Topology and its Applications

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We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes. We use this realization to unite the approach to A n -maps of Iwase and Mimura to that of Boardman and Vogt. We include a review of the appearance of the nth multiplihedron for various n in the studies of higher homotopy commutativity, (weak) n-categories, A ∞ -categories, deformation theory, and moduli spaces. We also include suggestions for the use of our realizations in some of these areas as well as in related studies, including enriched category theory and the graph-associahedra.

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“…However the permutohedron is also constructible by truncation of the cube, yielding X . This construction is not original; for example Devadoss and Forcey [4] use this truncation of the cube to construct the permutohedron.…”

Section: Toric Blowups and The Permutohedronmentioning

confidence: 99%