2,925 research outputs found
Combinatorial Seifert fibred spaces with transitive cyclic automorphism group
In combinatorial topology we aim to triangulate manifolds such that their
topological properties are reflected in the combinatorial structure of their
description. Here, we give a combinatorial criterion on when exactly
triangulations of 3-manifolds with transitive cyclic symmetry can be
generalised to an infinite family of such triangulations with similarly strong
combinatorial properties. In particular, we construct triangulations of Seifert
fibred spaces with transitive cyclic symmetry where the symmetry preserves the
fibres and acts non-trivially on the homology of the spaces. The triangulations
include the Brieskorn homology spheres , the lens spaces
and, as a limit case, .Comment: 28 pages, 9 figures. Minor update. To appear in Israel Journal of
Mathematic
Special Lagrangian torus fibrations of complete intersection Calabi-Yau manifolds: a geometric conjecture
For complete intersection Calabi-Yau manifolds in toric varieties, Gross and
Haase-Zharkov have given a conjectural combinatorial description of the special
Lagrangian torus fibrations whose existence was predicted by Strominger, Yau
and Zaslow. We present a geometric version of this construction, generalizing
an earlier conjecture of the first author.Comment: 23 pagers, 10 figure
Partitioning the triangles of the cross polytope into surfaces
We present a constructive proof that there exists a decomposition of the
2-skeleton of the k-dimensional cross polytope into closed surfaces
of genus , each with a transitive automorphism group given by the
vertex transitive -action on . Furthermore we show
that for each the 2-skeleton of the (k-1)-simplex is a union
of highly symmetric tori and M\"obius strips.Comment: 13 pages, 1 figure. Minor update. Journal-ref: Beitr. Algebra Geom. /
Contributions to Algebra and Geometry, 53(2):473-486, 201
On cyclic branched coverings of prime knots
We prove that a prime knot K is not determined by its p-fold cyclic branched
cover for at most two odd primes p. Moreover, we show that for a given odd
prime p, the p-fold cyclic branched cover of a prime knot K is the p-fold
cyclic branched cover of at most one more knot K' non equivalent to K. To prove
the main theorem, a result concerning the symmetries of knots is also obtained.
This latter result can be interpreted as a characterisation of the trivial
knot.Comment: 28 pages, 2 figure
Triangulated Manifolds with Few Vertices: Centrally Symmetric Spheres and Products of Spheres
The aim of this paper is to give a survey of the known results concerning
centrally symmetric polytopes, spheres, and manifolds. We further enumerate
nearly neighborly centrally symmetric spheres and centrally symmetric products
of spheres with dihedral or cyclic symmetry on few vertices, and we present an
infinite series of vertex-transitive nearly neighborly centrally symmetric
3-spheres.Comment: 26 pages, 8 figure
Diversity in the Tail of the Intersecting Brane Landscape
Techniques are developed for exploring the complete space of intersecting
brane models on an orientifold. The classification of all solutions for the
widely-studied T^6/Z_2 x Z_2 orientifold is made possible by computing all
combinations of branes with negative tadpole contributions. This provides the
necessary information to systematically and efficiently identify all models in
this class with specific characteristics. In particular, all ways in which a
desired group G can be realized by a system of intersecting branes can be
enumerated in polynomial time. We identify all distinct brane realizations of
the gauge groups SU(3) x SU(2) and SU(3) x SU(2) x U(1) which can be embedded
in any model which is compatible with the tadpole and SUSY constraints. We
compute the distribution of the number of generations of "quarks" and find that
3 is neither suppressed nor particularly enhanced compared to other odd
generation numbers. The overall distribution of models is found to have a long
tail. Despite disproportionate suppression of models in the tail by K-theory
constraints, the tail in the distribution contains much of the diversity of
low-energy physics structure.Comment: 48 pages, 8 figure
Many projectively unique polytopes
We construct an infinite family of 4-polytopes whose realization spaces have
dimension smaller or equal to 96. This in particular settles a problem going
back to Legendre and Steinitz: whether and how the dimension of the realization
space of a polytope is determined/bounded by its f-vector.
From this, we derive an infinite family of combinatorially distinct
69-dimensional polytopes whose realization is unique up to projective
transformation. This answers a problem posed by Perles and Shephard in the
sixties. Moreover, our methods naturally lead to several interesting classes of
projectively unique polytopes, among them projectively unique polytopes
inscribed to the sphere.
The proofs rely on a novel construction technique for polytopes based on
solving Cauchy problems for discrete conjugate nets in S^d, a new
Alexandrov--van Heijenoort Theorem for manifolds with boundary and a
generalization of Lawrence's extension technique for point configurations.Comment: 44 pages, 18 figures; to appear in Invent. mat
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