Computing Homology Generators for Volumes Using Minimal Generalized Maps

International audienceIn this paper, we present an algorithm for computing efficiently homology generators of 3D subdivided orientable objects which can contain tunnels and cavities. Starting with an initial subdivision, represented with a generalized map where every cell is a topological ball, the number of cells is reduced using simplification operations (removal of cells), while preserving homology. We obtain a minimal representation which is homologous to the initial object. A set of homology generators is then directly deduced on the simplified 3D object

Open Issues and Chances for Topological Pyramids

High resolution image data require a huge amount of computational resources. Image pyramids have shown high performance and flexibility to reduce the amount of data while preserving the most relevant pieces of information, and still allowing fast access to those data that have been considered less important before. They are able to preserve an existing topological structure (Euler number, homology generators) when the spatial partitioning of the data is known at the time of construction. In order to focus on the topological aspects let us call this class of pyramids “topological pyramids”. We consider here four open problems, under the topological pyramids context: The minimality problem of volumes representation, the “contact”-relation representation, the orientation of gravity and time dimensions and the integration of different modalities as different topologies.Austrian Science Fund P20134-N13Junta de Andalucía FQM–296Junta de Andalucía PO6-TIC-0226

A Tool for Integer Homology Computation: Lambda-At Model

In this paper, we formalize the notion of lambda-AT-model (where $\lambda$ is a non-null integer) for a given chain complex, which allows the computation of homological information in the integer domain avoiding using the Smith Normal Form of the boundary matrices. We present an algorithm for computing such a model, obtaining Betti numbers, the prime numbers p involved in the invariant factors of the torsion subgroup of homology, the amount of invariant factors that are a power of p and a set of representative cycles of generators of homology mod p, for each p. Moreover, we establish the minimum valid lambda for such a construction, what cuts down the computational costs related to the torsion subgroup. The tools described here are useful to determine topological information of nD structured objects such as simplicial, cubical or simploidal complexes and are applicable to extract such an information from digital pictures.Comment: Journal Image and Vision Computing, Volume 27 Issue 7, June, 200

Numeric Invariants from Multidimensional Persistence

We extend the results of Adcock, Carlsson, and Carlsson by constructing numeric invariants from the computation of a multidimensional persistence module as given by Carlsson, Singh, and Zomorodian.Comment: v1. initial upload. v2. fixed typos and rephrased sentence in introduction. v3. updated parameterization of rectangular persistence module

Heegaard Floer homology of L-space links with two components

We compute different versions of link Floer homology $HFL^{-}$ and $\widehat{HFL}$ for any $L$-space link with two components. The main approach is to compute the $h$-function of the filtered chain complex which is determined by the Alexander polynomials of every sublink of the $L$-space link. As an application, Thurston polytope and Thurston norm of any 2-component $L$-space link are explicitly determined by Alexander polynomials of the link and the link components.Comment: 23 page

Counting BPS Baryonic Operators in CFTs with Sasaki-Einstein duals

We study supersymmetric D3 brane configurations wrapping internal cycles of type II backgrounds AdS(5) x H for a generic Sasaki-Einstein manifold H. These configurations correspond to BPS baryonic operators in the dual quiver gauge theory. In each sector with given baryonic charge, we write explicit partition functions counting all the BPS operators according to their flavor and R-charge. We also show how to extract geometrical information about H from the partition functions; in particular, we give general formulae for computing volumes of three cycles in H.Comment: 46 pages, 10 figures; comments and clarifications added, published versio