537 research outputs found
Combinatorial 3-manifolds with 10 vertices
We give a complete enumeration of all combinatorial 3-manifolds with 10
vertices: There are precisely 247882 triangulated 3-spheres with 10 vertices as
well as 518 vertex-minimal triangulations of the sphere product
and 615 triangulations of the twisted sphere product S^2_\times_S^1.
All the 3-spheres with up to 10 vertices are shellable, but there are 29
vertex-minimal non-shellable 3-balls with 9 vertices.Comment: 9 pages, minor revisions, to appear in Beitr. Algebra Geo
Balanced triangulations on few vertices and an implementation of cross-fips
A
d
-dimensional simplicial complex is balanced if the underlying graph is
(
d
+
1
)
-colorable. We present an implementation of cross-flips, a set of local moves
introduced by Izmestiev, Klee and Novik which connect any two PL-homeomorphic
balanced combinatorial manifolds. As a result we exhibit a vertex minimal balanced
triangulation of the real projective plane, of the dunce hat and of the real projective
space, as well as several balanced triangulations of surfaces and 3-manifolds on few
vertices. In particular we construct small balanced triangulations of the 3-sphere
that are non-shellable and shellable but not vertex decomposable
Small examples of non-constructible simplicial balls and spheres
We construct non-constructible simplicial -spheres with vertices
and non-constructible, non-realizable simplicial -balls with vertices
for .Comment: 9 pages, 3 figure
Hyperplane Neural Codes and the Polar Complex
Hyperplane codes are a class of convex codes that arise as the output of a
one layer feed-forward neural network. Here we establish several natural
properties of stable hyperplane codes in terms of the {\it polar complex} of
the code, a simplicial complex associated to any combinatorial code. We prove
that the polar complex of a stable hyperplane code is shellable and show that
most currently known properties of the hyperplane codes follow from the
shellability of the appropriate polar complex.Comment: 23 pages, 5 figures. To appear in Proceedings of the Abel Symposiu
On -stellated and -stacked spheres
We introduce the class of -stellated (combinatorial) spheres
of dimension () and compare and contrast it with the
class () of -stacked homology -spheres.
We have , and for . However, for each there are
-stacked spheres which are not -stellated. The existence of -stellated
spheres which are not -stacked remains an open question.
We also consider the class (and ) of
simplicial complexes all whose vertex-links belong to
(respectively, ). Thus, for , while . Let
denote the class of -dimensional complexes all whose
vertex-links are -stacked balls. We show that for , there is a
natural bijection from onto which is the inverse to the boundary map .Comment: Revised Version. Theorem 2.24 is new. 18 pages. arXiv admin note:
substantial text overlap with arXiv:1102.085
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