1,720,974 research outputs found
Balanced triangulations on few vertices and an implementation of cross-fips
Full text linkA
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
Octahedralizing 3-Colorable 3-Polytopes
No full textWe investigate the question of whether any d-colorable simplicial d-polytope can be octahedralized, i.e., can be subdivided to a d-dimensional geometric cross-polytopal complex. We give a positive answer in dimension 3, with the additional property that the octahedralization introduces no new vertices on the boundary of the polytope
A New Family of Triangulations of RPd
No full textWe construct a family of PL triangulations of the d-dimensional real projective space P-d on vertices for every . This improves a construction due to Kuhnel on 2(d+1) -1 vertices
A balanced non-partitionable cohen-macaulay complex
Full text linkIn a recent article, Duval, Goeckner, Klivans and Martin disproved the longstanding conjecture by Stanley, that every Cohen–Macaulay simplicial complex is partitionable. We construct counterexamples to this conjecture that are even balanced, i.e. their underlying graph has a minimal coloring. This answers a question by Duval et al. in the negative
Balanced shellings and moves on balanced manifolds
Full text linkA classical result by Pachner states that two d-dimensional combinatorial manifolds with boundary are PL homeomorphic if and only if they can be connected by a sequence of shellings and inverse shellings. We prove that for balanced, i.e., properly (d + 1)-colored, manifolds such a sequence can be chosen such that balancedness is preserved in each step. As a key ingredient we establish that any two balanced PL homeomorphic combinatorial manifolds with the same boundary are connected by a sequence of basic cross-flips, as was shown recently by Izmestiev, Klee and Novik for balanced manifolds without boundary. Moreover, we enumerate combinatorially different basic cross-flips and show that roughly half of these suffice to relate any two PL homeomorphic manifolds. (C) 2021 Elsevier Inc. All rights reserved
Graded Betti Numbers of Balanced Simplicial Complexes
Full text linkWe prove upper bounds for the graded Betti numbers of Stanley-Reisner rings of balanced simplicial complexes. Along the way we show bounds for Cohen-Macaulay graded rings S/I, where S is a polynomial ring and I subset of S is a homogeneous ideal containing a certain number of generators in degree 2, including the squares of the variables. Using similar techniques we provide upper bounds for the number of linear syzygies for Stanley-Reisner rings of balanced normal pseudomanifolds. Moreover, we compute explicitly the graded Betti numbers of cross-polytopal stacked spheres, and show that they only depend on the dimension and the number of vertices, rather than also the combinatorial type
Balanced triangulations on few vertices and an implementation of cross-flips
Full text linkA 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 without boundary. As a result we exhibit a vertexminimal balanced triangulation of the dunce hat and balanced triangulations of several surfaces and 3-manifolds on few vertices. In particular we obtain small balanced triangulations of the 3-sphere that are non-shellable or shellable but not vertex decomposable
Spectral theory of weighted hypergraphs via tensors
No full textOne way to study a hypergraph is to attach to it a tensor. Tensors are a generalization of matrices, and they are an efficient way to encode information in a compact form. In this paper, we study how properties of weighted hypergraphs are reflected on eigenvalues and eigenvectors of their associated tensors. We also show how to efficiently compute eigenvalues with some techniques from numerical algebraic geometry
Eigenschemes of Ternary Tensors
No full textWe study projective schemes arising from eigenvectors of tensors, called eigenschemes. After some general results, we give a birational description of the variety parametrizing eigenschemes of general ternary symmetric tensors, and we compute its dimension. Moreover, we characterize the locus of triples of homogeneous polynomials defining the eigenscheme of a ternary symmetric tensor. Our results allow us to implement algorithms to check whether a given set of points is the eigenscheme of a symmetric tensor and to reconstruct the tensor. Finally, we give a geometric characterization of all reduced zero-dimensional eigenschemes. The techniques we use rely on both classical and modern complex projective algebraic geometry
Equations of tensor Eigenschemes
No full textWe study schemes of tensor eigenvectors from an algebraic and geometric viewpoint. We characterize determinantal defining equations of such eigenschemes via linear equations in their coefficients, both in the general and in the symmetric case. We give a geometric necessary condition for a 0-dimensional scheme to be an eigenscheme
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