Distributed computation of low-dimensional cup products

Alokbi, Nisreen; Ellis, Graham

doi:10.4310/hha.2018.v20.n2.a3

Homology, Homotopy and Applications

2018

DOI: 10.4310/hha.2018.v20.n2.a3

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Distributed computation of low-dimensional cup products

Nisreen Alokbi

Graham Ellis

Abstract: We describe a distributed algorithm for computing the cup product ∪ : H 1 (X, Z) × H 1 (X, Z) → H 2 (X, Z) on the cohomology of a finite regular CW-space. A serial implementation of the algorithm is illustrated in two applied topological settings: (i) 3-dimensional digital images; (ii) topological data analysis of a finite sample of points from a metric space. For the second of these illustrations we introduce a cohomological enrichment of the Mapper clustering procedure which may be of independent interest.

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Cited by 3 publications

References 21 publications

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Presentation of the subgroups of Mathieu Group using Groupoid

Alokbi

Mayah

2020

J. Phys.: Conf. Ser.

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Mathieu groups are one type of the sporadic simple groups, they turn out not to be isomorphic to any member of the infinite families of finite simple groups. Study these groups is interesting since their orders are very high. Groupoid can be used to find the presentation of the subgroups of the Mathieu groups. The idea is creating a groupoid by acting the Mathieu group on a subset of this group and then calculating the presentation of the vertex group of the groupoid which represents the presentation of the subgroup as the vertex groups are isomorphic.

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Presentation of the subgroups of Mathieu Group using Groupoid

Alokbi

Mayah

2020

J. Phys.: Conf. Ser.

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Cellular approximations to the diagonal map

Alzobydi,

Ellis

2024

Math. Comp.

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We describe an elementary algorithm for recursively constructing diagonal approximations on those finite regular CW-complexes for which the closure of each cell can be explicitly collapsed to a point. The algorithm is based on the standard proof of the acyclic carrier theorem, made constructive through the use of explicit contracting homotopies. It can be used as a theoretical tool for constructing diagonal approximations on families of polytopes in situations where the diagonals are required to satisfy certain coherence conditions. We compare its output to existing diagonal approximations for the families of simplices, cubes, associahedra and permutahedra. The algorithm yields a new explanation of a magical formula for the associahedron derived by Markl and Shnider [Trans. Amer. Math. Soc. 358 (2006), pp. 2353–2372] and Masuda, Thomas, Tonks, and Vallette [J. Éc. polytech. Math. 8 (2021), pp. 121–146] and Theorem 4.1 provides a magical formula for other polytopes. We also describe a computer implementation of the algorithm and illustrate it on a range of practical examples including the computation of cohomology rings for some low-dimensional manifolds. To achieve some of these examples the paper includes two approaches to generating a regular CW-complex structure on closed compact 3 3 -manifolds, one using an implementation of Dehn surgery on links and the other using an implementation of pairwise identifications of faces in a tessellated boundary of the 3 3 -ball. The latter is illustrated in Proposition 8.1 with a topological classification of all closed orientable 3 3 -manifolds arising from pairwise identifications of faces of the cube.

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New Invariant Quantity To Measure The Entanglement In The Braids

Mayah

Alokbi

Rasheed

2022

J. Nig. Soc. Phys. Sci.

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In this work, we demonstrate that the integral formula for a generalised Sato-Levine invariant is consistent in certain situations with Evans and Berger's formula for the fourth-order winding number. Also, we found that, in principle, one can derive analogous high-order winding numbers by which one can calculate the entanglement of braids. The winding number for the Brunnian 4-braid is calculated algebraically using the cup product on the cohomology of a finite regular CW-space which is the complement $\mathbb{R}^3\backslash \mathcal{B}_4$.

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Distributed computation of low-dimensional cup products

Cited by 3 publications

References 21 publications

Presentation of the subgroups of Mathieu Group using Groupoid

Presentation of the subgroups of Mathieu Group using Groupoid

Cellular approximations to the diagonal map

New Invariant Quantity To Measure The Entanglement In The Braids

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