Cubic Manifolds in Lattices

Долбилин, Николай Петрович; Shtan'ko, M A; Штогрин, Михаил Иванович

doi:10.1070/im1995v044n02abeh001598

Cited by 21 publications

(29 citation statements)

References 5 publications

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“…In Section 2, we give necessary preliminary material about simplicial and cubical complexes and their homology properties, following [6], [10], and [12]. In particular, we discuss in detail the procedure for constructing of the cubical complex Q S mentioned above.…”

Section: Introductionmentioning

confidence: 99%

Graphs associated with simplicial complexes

Grigor’yan

Muranov

Yau

2014

Homology, Homotopy and Applications

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The cohomology of digraphs was introduced for the first time by Dimakis and Müller-Hoissen. Their algebraic definition is based on a differential calculus on an algebra of functions on the set of vertices with relations that follow naturally from the structure of the set of edges. A dual notion of homology of digraphs, based on the notion of path complex, was introduced by the authors, and the first methods for computing the (co)homology groups were developed. The interest in homology on digraphs is motivated by physical applications and relations between algebraic and geometrical properties of quivers. The digraph G B of the partially ordered set B S of simplexes of a simplicial complex S has graph homology that is isomorphic to the simplicial homology of S. In this paper, we introduce the concept of cubical digraphs and describe their homology properties. In particular, we define a cubical subgraph G S of G B , whose homologies are isomorphic to the simplicial homologies of S.

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Section: Introductionmentioning

confidence: 99%

Graphs associated with simplicial complexes

Grigor’yan

Muranov

Yau

2014

Homology, Homotopy and Applications

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“…Such a construction is already known and studied (see, for instance, Shtan 0 ko and Shtogrin [15], Dolbilin and Shtan 0 ko and Shtogrin [4] and Funar [5]), but we have not found any application to nullhomotopic filling Dehn spheres in the literature.…”

Section: Introductionmentioning

confidence: 99%

A local calculus for nullhomotopic filling Dehn spheres

Amendola

2009

Algebr. Geom. Topol.

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We provide a local calculus for the presentation of closed 3-manifolds via nullhomotopic filling Dehn spheres. We use it to define an invariant of closed 3-manifolds by applying the state-sum machinery, and we show how to potentially get lower bounds for the Matveev complexity of P 2 -irreducible closed 3-manifolds. We also describe an efficient and simple algorithm for constructing a nullhomotopic filling Dehn sphere of each closed 3-manifold from any of its one-vertex triangulations.

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“…We start with a cubical octahedron O 8 , the smallest cubical 3-polytope that is not a cube, with f -vector f (O 8 ) = (10,16,8). The special feature of this complex is that it contains a cubical Möbius strip with parallel inner edges of length 9 in its 2-skeleton, as is illustrated in the figure.…”

Section: 2])mentioning

confidence: 99%

“…[8]). In particular, for each k-face F of M d−1 the image j(F ) is a k-face of the skeleton of the standard unit cube structure on R d .…”

Section: Cubical 4-polytopes With Prescribed Dual Manifold Immersionsmentioning

confidence: 99%

Construction Techniques for Cubical Complexes, Odd Cubical 4-Polytopes, and Prescribed Dual Manifolds

Schwartz

Ziegler

2004

Experimental Mathematics

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We provide a number of new construction techniques for cubical complexes and cubical polytopes, and thus for cubifications (hexahedral mesh generation). Thus we obtain the first instance of a cubical 4-polytope that has a non-orientable dual manifold (a Klein bottle). This confirms the existence conjecture of Hetyei [17, Conj. 2, p. 325]. More systematically, we prove that every normal crossing codimension one immersion of a compact 2-manifold into R 3 is PL-equivalent to a dual manifold immersion of a cubical 4-polytope. As an instance we obtain a cubical 4-polytope with a cubation of Boy's surface as a dual manifold immersion, and with an odd number of facets. This solves problems of Eppstein, Thurston and others. Our explicit example has 19 520 vertices and 18 333 facets.

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Cubic Manifolds in Lattices

Cited by 21 publications

References 5 publications

Graphs associated with simplicial complexes

Graphs associated with simplicial complexes

A local calculus for nullhomotopic filling Dehn spheres

Construction Techniques for Cubical Complexes, Odd Cubical 4-Polytopes, and Prescribed Dual Manifolds

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