Data structures and algorithms for topological analysis

Set operation evaluation is an important component of Geometric Modeling System (GMS). Input data can be a CSG (Constructive Solid Geometry) representation with solid primitives approximated by polyhedra. This representation is a powerful tool for solid object description. Output data will be the unique resulting polyhedral object which provides an efficient data structure in display field. With no use of spatial coherency, computational complexity of set operation is quadratic. In this paper, we are introducing a new space subdivision scheme called Boolean Octree which performs set operations on polyhedra in an efficient way. This structure aims at limiting set operation evaluation in a ‘minimal space of calculation’ where primitive boundaries intersect each other and where resulting evaluation participates in the construction of the final resulting object. Boolean Octree computes set operations in a local level providing a linear complexity for geometric calculations. During space subdivision, Boolean Octree has a global view on local CSG tree (projection of the CSG tree in local space) taking into account simplifications of the boolean expression, avoiding evaluation and subdivision for the object parts out of the ‘minimal space of calculation’.

The main goal of this chapter is to provide an introduction to simplicial and singular homology. We begin by stating the structure theorem for finitely abelian groups. Using this result, we define the Betti numbers. We state the fundamental fact that if two complexes have the same geometric realization, then these complexes and their geometric realization have the same homology groups. We also show how the Euler–Poincaré characteristic of a complex is given by the alternating sum of the Betti numbers of its homology groups. This shows that the Euler–Poincaré characteristic is an invariant of all the complexes corresponding to the same polytope. We determine the homology groups of surfaces with or without borders.

: Described is a set of algorithms for efficiently rendering a CSG- defined objective directly into a frame buffer without converting first to a boundary representation. The method requires only that the frame buffer contain sufficient memory to hold two color values, two z values, and three one-bit flags. The algorithm first converts the CSG tree to a normalized form that is analogous to the sum-of-products form for boolean switching functions. Expanding on results previously reported the authors, introduce 1) an improved algorithm for converting a CSG expression to sum-of-products form, 2) a technique for minimizing the tree using don't care conditions based on geometry, 3) the dynamic interleaving of the above two algorithms, allowing very efficient rendering for most CSG objects, 4) a method for extending the technique to non- convex primitives, and 5) an implementation of these ideas in an interactive CSG design system on Pixelplanes 4. In this system the designer directly manipulates the CSG structure while continuously viewing the color rendering of the object under design. It is believed these algorithms will attain similar speeds on many of the next generation high-performance graphics system that have frame buffers with many bits per pixel.

The generation of 3-D solid objects, and more generally solid geometric modelling, is very important in Computer Aided Design (CAD). An important role is played by the Constructive Solid Geometry (CSG) representation scheme. In CSG, objects are described by trees of Boolean operations on half-spaces or boundaries of primitive solids. The study of techniques to speed up the rendering of scenes modelled with the CSG scheme is an attractive field of research; in this paper we propose a new algorithm which reduces the computational complexity for ray casting approaches. Our strategy identifies a set of areas on the plane of view where the rays starting from the observer have to be traced; for each zone, only a portion of the entire CSG tree has to be considered for intersection tests, instead of the whole database of the primitive objects. A comparison of our algorithm with a ray caster that adopts bounding volume hierarchies and with a freeware ray tracer called POV-Ray shows that, for the examples considered, we may reduce the intersection tests to one third of those performed when standard optimizations are adopted.

Quandle homology groups, their Betti numbers, and virtual knots

This paper seeks to demonstrate the relation between homology group and homotopy group. The result in this paper is a construction of the homology of a complex torus.

In the high-voltage power grid of coal mine, the existing adaptive topology analysis algorithm based on the correlation matrix for coal mine high-voltage power grid can effectively identify the network topology and construct a network topology analysis model to provide a basis for subsequent short-circuit current calculation and setting calculation. However, the topology analysis algorithm based on the correlation matrix takes much time. As the scale of the high-voltage power grid network of the coal mine increases, the time consumption will increase sharply as well. However, the prior-to-first-served parallel topology analysis algorithm has been used to reduce the topology analysis time consumption to some extent. On this basis, this article proposes a parallel topology analysis algorithm based on the simulated annealing and optimizes the scheduling strategy of parallel topology analysis. The simulation results show that compared with the prior-to-first-served parallel topology analysis algorithm, the parallel annealing topology analysis algorithm can further reduce the time consumption and improve the computational efficiency. And when the number of available threads increases, its computational efficiency will improve more significantly.

This article delves into algebraic topology, specifically (co)homology theory, which is essential in various mathematical fields. It explores different types of (co)homology groups such as Hochschild, cyclic, reflexive, and dihedral, focusing on dihedral cohomology generated by the dihedral group operating on simplicial complexes. The text discusses when dihedral and reflexive cohomology vanishes and provides examples, along with proving a homomorphism between dihedral cohomology groups. Mathematical concepts like Banach spaces, C*-algebras, and tensor products are elucidated to understand the algebraic structures involved. Theorems and proofs establish the relationships and properties of these cohomology groups, culminating in a canonical isomorphism theorem for the product of two algebras. Overall, the article aims to provide a comprehensive exploration of dihedral cohomology and its interplay with other algebraic structures, offering insights into its applications and theoretical foundations. Additionally, it studies the (co)homology theory of C*-algebras, focusing on the triviality of cohomology groups of operator algebras and discovering the canonical isomorphism between any two unital K -algebras A and A′: HDn(A×A′)≅HDn(A)⊕HDn(A′).

In this article, we introduce a fixed parameter tractable algorithm for computing the Turaev-Viro invariants TV4,q, using the dimension of the first homology group of the manifold as parameter.This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parameter. The computation of TV4,q is known to be #P-hard in general; using a topological parameter provides an algorithm polynomial in the size of the input triangulation for the extremely large family of 3-manifolds with first homology group of bounded rank.Our algorithm is easy to implement and running times are comparable with running times to compute integral homology groups for standard libraries of triangulated 3-manifolds. The invariants we can compute this way are powerful: in combination with integral homology and using standard data sets we are able to roughly double the pairs of 3-manifolds we can distinguish.We hope this qualifies TV4,q to be added to the short list of standard properties (such as orientability, connectedness, Betti numbers, etc.) that can be computed ad-hoc when first investigating an unknown triangulation.

In this article, we introduce a fixed parameter tractable algorithm for computing the Turaev-Viro invariants TV(4,q), using the dimension of the first homology group of the manifold as parameter. This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parameter. The computation of TV(4,q) is known to be #P-hard in general; using a topological parameter provides an algorithm polynomial in the size of the input triangulation for the extremely large family of 3-manifolds with first homology group of bounded rank. Our algorithm is easy to implement and running times are comparable with running times to compute integral homology groups for standard libraries of triangulated 3-manifolds. The invariants we can compute this way are powerful: in combination with integral homology and using standard data sets we are able to roughly double the pairs of 3-manifolds we can distinguish. We hope this qualifies TV(4,q) to be added to the short list of standard properties (such as orientability, connectedness, Betti numbers, etc.) that can be computed ad-hoc when first investigating an unknown triangulation.

The Betti numbers of the rotation group, R(n), in n real variables were first determined by R. Brauer [5].2 Later Pontrjagin [12] extended slightly these results by computing the integral homology groups of R(n). The Betti numbers combined with Hopf's Theorem on group-manifolds [9], which states that the rational cohomology ring of a group-manifold is isomorphic to that of a product of odd dimensional spheres, can be used to determine abstractly the rational cohomology ring of R(n). Pontrjagin's results can be used to find the additive structure of the cohomology groups of R(n) over an arbitrary coefficient domain, but there are no general theorems such as Hopf's Theorem available for the determination of the cup product over coefficient rings other than the rational numbers. This paper gives a treatment of the cohomology structure of R(n) which links that structure more intimately with the space R(n) than does Pontrjagin's treatment and extends the results mentioned above to other coefficient domains. The basic tool used is the notion of a cell complex. The particular cell complex used on R(n) gives rise to a cell complex on the Stiefel manifold V(n, r) of r-frames in euclidian n-space, which J. H. C. Whitehead used [17] to compute some of the higher homotopy groups of V(n, r). In Section 2 we give a description of this cell complex, rephrasing Whitehead's definitions slightly in order to facilitate later computations. Sections 3, 5, and 7 are devoted to the computation of the homology and cohomology groups and the calculation of the cup product. In Section 6 the Steenrod squaring homomorphisms are determined for V(n, r). These are then used in Section 9 to give a proof of the Steenrod-Whitehead Theorem that 2k independent vector fields do not exist on S'-', where 2k is the largest power of 2 dividing n [16], and are also used in Section 11 to show that, unless 2n = 2-k 2, S2n has no quasi-complex structure.3 In each case the desired result is obtained by employing the squaring homomorphism to show the nonexistence of cross sections in certain fibre bundles. The appendix gives a brief account of a technique for finding the integral cup

This paper establishes sufficient conditions for the vanishing of the homology and cohomology groups of an associative algebra with coefficients in a two-sided module.

Computation of the homotopy groups π n ( X ) of a topological space X has played a central role in homotopy theory. And knowledge of these homotopy groups has inherent use and interest. Furthermore, the development of techniques to compute these groups has proven useful in many other contexts. The study of homotopy groups falls into three parts. First, there is the computation of specific homotopy groups π n ( X ) of spaces. This may be traced back to Poincare [106] in the case n = 1: Poincare : π 1 ( X )/[π 1 ( X ), π 1 ( X )] is isomorphic to H 1 ( X ). Hurewicz [62] showed that, in the simply connected case, the Hurewicz homomorphism provides an isomorphism of the first nonzero π n ( X ) with the homology group H n ( X ) with n ≥ 1: Hurewicz : If X is an n − 1 connected space with n ≥ 2, then π n ( X ) is isomorphic to H n ( X ). Hopf [58] discovered the remarkable fact that homotopy groups could be nonzero in dimensions higher than those of nonvanishing homology groups. He did this by using linking numbers but the modern way is to use the long exact sequence of the Hopf fibration sequence S 1 → S 3 → S 2 . Hopf : π 3 ( S 2 ) is isomorphic to the additive group of integers ℤ. Computation enters the modern era with the work of Serre [116, 118] on the low dimensional homotopy groups of spheres.

Consider an arrangement [Formula: see text] of homogeneous hyperplanes in [Formula: see text] with complement [Formula: see text]. The (co)homology of [Formula: see text] with twisted coefficients is strictly related to the cohomology of the Milnor fiber associated to the natural fibration onto [Formula: see text] endowed with the geometric monodromy. It is still an open problem to understand in general the cohomology of the Milnor fiber, even for dimension 1. In Sec. 1, we show that all questions about the first homology group are detected by a precise group, which is a quotient ot the commutator subgroup of [Formula: see text] by the commutator of its length zero subgroup, which didn’t appear in the literature before. In Sec. 2, we state a conjecture of [Formula: see text]-monodromicity for the first homology, which is of a different nature with respect to the known results. Let [Formula: see text] be the graph of double points of [Formula: see text] we conjecture that if [Formula: see text] is connected, then the geometric monodromy acts trivially on the first homology of the Milnor fiber (so the first Betti number is combinatorially determined in this case). This conjecture depends only on the combinatorics of [Formula: see text]. We show the truth of the conjecture under some stronger hypotheses.

We generalize Biggs Theorem to the case of directed cycles of multi-digraphs allowing to compute the dimension of the directed cycle space independently of the graph representation with linear runtime complexity. By considering two-dimensional CW complex of elementary cycles and deriving formulas for the Betti numbers of the associated cellular homology groups, we extend the list of representation independent topological inavariants measuring the graph structure. We prove the computation of the 2nd Betti number to be sharp #P hard in general and present specific representation invariant sub-fillings yielding efficiently computable homology groups. Finally, we suggest how to use the provided structural measures to shed new light on graph theoretical problems as graph embeddings, discrete Morse theory and graph clustering.