Computing Betti Numbers Research Articles

An approximation to Čech complex using median of triangles for computing Betti numbers of some point cloud data

An approach has been developed to create an approximated simplicial complex in between the Vietoris-Rips complex and the Čech complex using median of triangles for computing Betti numbers of some point cloud data. The Vietoris-Rips complex has been built first for this. Then the sample points have been classified into three classes based on three conditions of the median (l) of any triangle’s maximum edge (2r) for any three points in〖 R〗^n. Then the values of the filtration (ε) have been chosen in such a way that ε=r for l<r, ε=r for l=r, and ε=r+((l-r))/3 for l>r. The approach has been extended for higher dimensional triangles calculating l by the distance of the centroid from the opposite vertex of the maximum face and considering r as the filtration value of the maximum edge. Then an algorithm has been introduced to calculate the simplicial complex after building simplices for each filtration value. Finally, to validate the study results of the approximated simplicial complex have been compared with the Vietoris-Rips complex and the Čech complex. The proposed approximated simplicial complex has been found computationally effective than the Čech complex and its filtration values are lying between filtration values of the Vietoris-Rips complex and the Čech complex without any loss of persistent data.

When Is a Graded Free Complex Exact?

Minimal free resolutions of a finitely generated module over a polynomial ring S=k[x], with variables x={x1,…,xn} and a field k have been extensively studied. Almost all the results in the literature about minimal free resolutions give their Betti numbers, that is, the ranks of the free modules in the resolution at each degree. Several techniques have been developed to compute Betti numbers, making this a manageable problem in many cases. However, a description of the differentials in the resolution is rarely given, as this turns out to be a more difficult problem. The main purpose of this article is to give a criterion to check when a graded free complex of an S-module is exact. Unlike previous similar criteria, this one allows us to give a description of the differentials using the combinatorics of the S-module. The criterion is given in terms of the Betti numbers of the resolutions in each degree and the set of columns of the matrix representation of the differentials. In the last section, and with the aim of illustrating how to use the criterion, we apply it to one of the first better-understood cases, the edge ideal of the complete graph. However, this criterion can be used to give an explicit description of the differentials of a resolution of several monomial ideals such as the duplication of an ideal, the edge ideal of a cograph, etc.

English

Let o(n) be the greatest odd integer less than or equal to n. In this paper we provide explicit formulae to compute ℕ-graded Betti numbers of the circulant graphs C2n(1, 2, 3, 5, …, o(n)). We do this by showing that this graph is the product (or join) of the cycle Cn by itself, and computing Betti numbers of Cn * Cn. We also discuss whether such a graph (more generally, G * H) is well-covered, Cohen-Macaulay, sequentially Cohen-Macaulay, Buchsbaum, or S2.

BETTI NUMBERS FOR CERTAIN COHEN–MACAULAY TANGENT CONES

We compute Betti numbers for a Cohen–Macaulay tangent cone of a monomial curve in the affine $4$-space corresponding to a pseudo-symmetric numerical semigroup. As a byproduct, we also show that for these semigroups, being of homogeneous type and homogeneous are equivalent properties.

Intersection Poincaré Polynomial for Nagaraj–Seshadri moduli space

AbstractWe compute Betti numbers of both the components of the moduli space of rank 2 semi‐stable torsion‐free sheaves with fixed determinant over a reducible nodal curve with two smooth components intersecting at a node. We also compute the intersection Betti numbers of the moduli space.

The topology of the cosmic web in terms of persistent Betti numbers

We introduce a multiscale topological description of the Megaparsec weblike cosmic matter distribution. Betti numbers and topological persistence offer a powerful means of describing the rich connectivity structure of the cosmic web and of its multiscale arrangement of matter and galaxies. Emanating from algebraic topology and Morse theory, Betti numbers and persistence diagrams represent an extension and deepening of the cosmologically familiar topological genus measure, and the related geometric Minkowski functionals. In addition to a description of the mathematical background, this study presents the computational procedure for computing Betti numbers and persistence diagrams for density field filtrations. The field may be computed starting from a discrete spatial distribution of galaxies or simulation particles. The main emphasis of this study concerns an extensive and systematic exploration of the imprint of different weblike morphologies and different levels of multiscale clustering in the corresponding computed Betti numbers and persistence diagrams. To this end, we use Voronoi clustering models as templates for a rich variety of weblike configurations, and the fractal-like Soneira-Peebles models exemplify a range of multiscale configurations. We have identified the clear imprint of cluster nodes, filaments, walls, and voids in persistence diagrams, along with that of the nested hierarchy of structures in multiscale point distributions. We conclude by outlining the potential of persistent topology for understanding the connectivity structure of the cosmic web, in large simulations of cosmic structure formation and in the challenging context of the observed galaxy distribution in large galaxy surveys.

Resolutions and Betti Numbers of Polynomial Modules via Involutive Bases

AbstractWe describe a novel approach to the computation of free resolutions and of Betti numbers of polynomial modules based on a combination of the theory of involutive bases with algebraic discrete Morse theory. This approach allows for the first time to compute Betti numbers (even single ones) without determining a whole resolution which in many cases drastically reduces the computation time. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Relative Hilbert scheme of points

Let $D$ be a smooth divisor on a non singular surface $S$. We compute Betti numbers of the relative Hilbert scheme of points of $S$ relative to $D$. In the case of $\PP^2$ and a line in it, we give an explicit set of generators and relations for the cohomology groups of this space.

Polynomial Hierarchy, Betti Numbers, and a Real Analogue of Toda’s Theorem

Toda (in SIAM J. Comput. 20(5):865–877, 1991) proved in 1989 that the (discrete) polynomial time hierarchy, PH, is contained in the class P #P , namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class #P. This result, which illustrates the power of counting, is considered to be a seminal result in computational complexity theory. An analogous result in the complexity theory over the reals (in the sense of Blum–Shub–Smale real machines in Bull. Am. Math. Soc. (NS) 21(1): 1–46, 1989) has been missing so far. In this paper we formulate and prove a real analogue of Toda’s theorem. Unlike Toda’s proof in the discrete case, which relied on sophisticated combinatorial arguments, our proof is topological in nature. As a consequence of our techniques, we are also able to relate the computational hardness of two extremely well-studied problems in algorithmic semi-algebraic geometry: the problem of deciding sentences in the first-order theory of the reals with a constant number of quantifier alternations, and that of computing Betti numbers of semi-algebraic sets. We obtain a polynomial time reduction of the compact version of the first problem to the second. This latter result may be of independent interest to researchers in algorithmic semi-algebraic geometry.

On the complexity of deciding connectedness and computing Betti numbers of a complex algebraic variety

We extend the lower bounds on the complexity of computing Betti numbers proved in [P. Bürgisser, F. Cucker, Counting complexity classes for numeric computations II: algebraic and semialgebraic sets, J. Complexity 22 (2006) 147–191] to complex algebraic varieties. More precisely, we first prove that the problem of deciding connectedness of a complex affine or projective variety given as the zero set of integer polynomials is PSPACE-hard. Then we prove PSPACE-hardness for the more general problem of deciding whether the Betti number of fixed order of a complex affine or projective variety is at most some given integer.

Computing Betti Numbers via Combinatorial Laplacians

We use the Laplacian and power method to compute Betti numbers of simplicial complexes. This has a number of advantages over other methods, both in theory and in practice. It requires small storage space in many cases. It seems to run quickly in practice, but its running time depends on a ratio, ν , of eigenvalues which we have yet to understand fully. We numerically verify a conjecture of Bj{o}rner, Lov{a}sz, Vre{\'c}ica, and {\u Z}ivaljevi{\'c} on the chessboard complexes C(4,6) , C(5,7) , and C(5,8) . Our verification suffers a technical weakness, which can be overcome in various ways; we do so for C(4,6) and C(5,8) , giving a completely rigorous (computer) proof of the conjecture in these two cases. This brings up an interesting question in recovering an integral basis from a real basis of vectors.

The cohomology rings of moduli spaces of bundles over Riemann surfaces

The cohomology of the moduli space JI(n, d) of semistable bundles of coprime rank n and degree d over a fixed Riemann surface M of genus g > 2 has been much studied over the last two decades, and a great deal is known about it. It is known that the cohomology has no torsion [2], the Betti numbers have been computed using various different methods [2, 5, 11], and a set of generators for the cohomology ring is known [2]. The purpose of this article is to provide a method of deciding when an expression in these generators is a relation, and in particular, when the rank n is two, to prove two conjectures about the relations between the generators due to Mumford and to Newstead and Ramanan. Newstead and Ramanan conjectured that certain relations hold among the generators, whereas Mumford conjectured that a certain set of relations is a complete set. In fact the conjecture of Newstead and Ramanan has two parts, the first of which was proved by Gieseker in 1982 [4]. The second part of their conjecture, which is the part examined in this article, was given an independent proof recently by Thaddeus [19], who has a very different method of obtaining information about the relations in the cohomology ring. The computation of the Betti numbers of the moduli spaces .I'(n, d) in [2] stimulated a general method of computing Betti numbers of quotients in the sense of Mumford's geometric invariant theory [14] of nonsingular complex projective varieties by complex reductive group actions (see [8, 9, 10]). Similarly the ideas applied in this article to finding generators and relations for the cohomology rings of the moduli spaces .I'(n, d) also lead to an analogous method of obtaining generators and relations for the rational cohomology rings of geometric invariant theoretic quotients. The first section provides background material on the cohomology of the moduli spaces .I'(n, d) . In the second section, Mumford's conjecture is proved when the rank n is two. The method could, in principle, be used to decide whether the conjecture is true for rank n > 2 but appears to become significantly more complicated in the general case. The third section gives a method for deciding when an expression in the generators is a relation, and the fourth

Computation of cohomology operations of finite simplicial complexes

We propose a method for calculating cohomology operations on finite simplicial complexes. Of course, there exist well‐known methods for computing (co)homology groups, for example, the “reduction algorithm” consisting in reducing the matrices corresponding to the differential in each dimension to the Smith normal form, from which one can read o the (co)homology groups of the complex [Mun84], or the “incremental algorithm” for computing Betti numbers [DE93]. Nevertheless, little is known about general methods for computing cohomology operations. For any finite simplicial complex K, we give a procedure including the computation of some primary and secondary cohomology operations. This method is based on the transcription of the reduction algorithm mentioned above, in terms of a special type of algebraic homotopy equivalences, called contractions [McL75], of the (co)chain complex of K to a “minimal” (co)chain complex M(K). More concretely, whenever the ground ring is a field or the (co)homology of K is free, then M(K) is isomorphic to the (co)homology of K. Combining this contraction with the combinatorial formulae for Steenrod reduced pth powers at cochain level developed in [GR99] and [Gon00], these operations at cohomology level can be computed. Finally, a method for calculating Adem secondary cohomology operations 'q : Ker(Sq 2 H q (K)) ! H q+3 (K)=Sq 2 H q (K) is showed.