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17
MARKED TUBES AND THE GRAPH MULTIPLIHEDRON
, 2008
"... Given a graph G, we construct a convex polytope whose face poset is based on marked subgraphs of G. Dubbed the graph multiplihedron, we provide a realization using integer coordinates. Not only does this yield a natural generalization of the multiphihedron, but features of this polytope appear in w ..."
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Cited by 11 (2 self)
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Given a graph G, we construct a convex polytope whose face poset is based on marked subgraphs of G. Dubbed the graph multiplihedron, we provide a realization using integer coordinates. Not only does this yield a natural generalization of the multiphihedron, but features of this polytope appear in works related to quilted disks, bordered Riemann surfaces, and operadic structures. Certain examples of graph multiplihedra are related to Minkowski sums of simplices and cubes and others to the permutohedron.
Quotients of the multiplihedron as categorified associahedra
 HOMOTOPY, HOMOLOGY AND APPL
, 2008
"... We describe a new sequence of polytopes which characterize A ∞ maps from a topological monoid to an A ∞ space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Our sequence of polytopes is demonstrated not to be combinatorially equivalent to the associahedra, as w ..."
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Cited by 10 (4 self)
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We describe a new sequence of polytopes which characterize A ∞ maps from a topological monoid to an A ∞ space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Our sequence of polytopes is demonstrated not to be combinatorially equivalent to the associahedra, as was previously assumed in both topological and categorical literature. They are given the new collective name composihedra. We point out how these polytopes are used to parameterize compositions in the formulation of the theories of enriched bicategories and pseudomonoids in a monoidal bicategory. We also present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence of
Geometric Realizations of the Multiplihedron and its complexification
"... We realize Stasheff’s multiplihedron geometrically as the moduli space of stable quilted disks. This generalizes the geometric realization of the associahedron as the moduli space of stable disks. We also construct an algebraic variety that has the multiplihedron as its nonnegative real part, and u ..."
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Cited by 9 (2 self)
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We realize Stasheff’s multiplihedron geometrically as the moduli space of stable quilted disks. This generalizes the geometric realization of the associahedron as the moduli space of stable disks. We also construct an algebraic variety that has the multiplihedron as its nonnegative real part, and use it to define a notion of morphism of cohomological field theories.
Hopf structures on the multiplihedra
, 2009
"... We investigate algebraic structures that can be placed on vertices of the multiplihedra, a family of polytopes originating in the study of higher categories and homotopy theory. Most compelling among these are two distinct structures of a Hopf module over the Loday–Ronco Hopf algebra. ..."
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Cited by 4 (3 self)
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We investigate algebraic structures that can be placed on vertices of the multiplihedra, a family of polytopes originating in the study of higher categories and homotopy theory. Most compelling among these are two distinct structures of a Hopf module over the Loday–Ronco Hopf algebra.
Lifted generalized permutahedra and composition polynomials
, 2012
"... We introduce a “lifting” construction for generalized permutohedra, which turns an ndimensional generalized permutahedron into an (n + 1)dimensional one. We prove that this construction gives rise to Stasheff’s multiplihedron from homotopy theory, and to the more general “nestomultiplihedra,” an ..."
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Cited by 2 (0 self)
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We introduce a “lifting” construction for generalized permutohedra, which turns an ndimensional generalized permutahedron into an (n + 1)dimensional one. We prove that this construction gives rise to Stasheff’s multiplihedron from homotopy theory, and to the more general “nestomultiplihedra,” answering two questions of Devadoss and Forcey. We construct a subdivision of any lifted generalized permutahedron whose pieces are indexed by compositions. The volume of each piece is given by a polynomial whose combinatorial properties we investigate. We show how this “composition polynomial ” arises naturally in the polynomial interpolation of an exponential function. We prove that its coefficients are positive integers, and conjecture that they are unimodal.
Associahedral categories, particles and Morse functor
, 2009
"... Every smooth manifold contains particles which propagate. These form objects and ..."
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Cited by 1 (0 self)
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Every smooth manifold contains particles which propagate. These form objects and