scholarly journals Homology groups of $\mathbb{C}{\Omega}_n$ space for certain dimensionalities

2016 ◽  
Vol 24 ◽  
pp. 71
Author(s):  
A.M. Pas'ko
Keyword(s):  
Euler Characteristic ◽  
Homology Groups

We calculate the homology groups $H_k(\mathbb{C}{\Omega}_n)$, $k = 0,1,2,2n-1,2n,2n+1$. We establish that the $\mathbb{C}{\Omega}_n$ space has a zero Euler characteristic.

scholarly journals Euler characteristics and their congruences in the positive rank setting

2020 ◽  
pp. 1-18
Author(s):  
Anwesh Ray ◽  
R. Sujatha
Keyword(s):  
Elliptic Curves ◽  
Euler Characteristic ◽  
Selmer Groups ◽  
Euler Characteristics ◽  
Rank Zero ◽  
Homology Groups ◽  
Main Conjecture ◽  
Congruence Relations ◽  
Higher Rank ◽  
Positive Rank

Abstract The notion of the truncated Euler characteristic for Iwasawa modules is an extension of the notion of the usual Euler characteristic to the case when the homology groups are not finite. This article explores congruence relations between the truncated Euler characteristics for dual Selmer groups of elliptic curves with isomorphic residual representations, over admissible p-adic Lie extensions. Our results extend earlier congruence results from the case of elliptic curves with rank zero to the case of higher rank elliptic curves. The results provide evidence for the p-adic Birch and Swinnerton-Dyer formula without assuming the main conjecture.

scholarly journals Khovanov Homology of Three-Strand Braid Links

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 720
Author(s):  
Young Kwun ◽  
Abdul Nizami ◽  
Mobeen Munir ◽  
Zaffar Iqbal ◽  
Dishya Arshad ◽  
...  
Keyword(s):  
Euler Characteristic ◽  
Jones Polynomial ◽  
Khovanov Homology ◽  
Homology Groups ◽  
Link Invariants ◽  
Chain Complexes ◽  
Chain Homotopy

Khovanov homology is a categorication of the Jones polynomial. It consists of graded chain complexes which, up to chain homotopy, are link invariants, and whose graded Euler characteristic is equal to the Jones polynomial of the link. In this article we give some Khovanov homology groups of 3-strand braid links Δ 2 k + 1 = x 1 2 k + 2 x 2 x 1 2 x 2 2 x 1 2 ⋯ x 2 2 x 1 2 x 1 2 , Δ 2 k + 1 x 2 , and Δ 2 k + 1 x 1 , where Δ is the Garside element x 1 x 2 x 1 , and which are three out of all six classes of the general braid x 1 x 2 x 1 x 2 ⋯ with n factors.

scholarly journals On the Loop Homology of a Certain Complex of RNA Structures

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1749
Author(s):  
Thomas J. X. Li ◽  
Christian M. Reidys
Keyword(s):  
Euler Characteristic ◽  
Homology Group ◽  
Secondary Structures ◽  
Rna Structures ◽  
Sequence Structure ◽  
Evolutionary Transitions ◽  
Partial Matching ◽  
Rna Sequence ◽  
Homology Groups ◽  
Topological Framework

In this paper, we establish a topological framework of τ-structures to quantify the evolutionary transitions between two RNA sequence–structure pairs. τ-structures developed here consist of a pair of RNA secondary structures together with a non-crossing partial matching between the two backbones. The loop complex of a τ-structure captures the intersections of loops in both secondary structures. We compute the loop homology of τ-structures. We show that only the zeroth, first and second homology groups are free. In particular, we prove that the rank of the second homology group equals the number γ of certain arc-components in a τ-structure and that the rank of the first homology is given by γ−χ+1, where χ is the Euler characteristic of the loop complex.

scholarly journals Matching and Independence Complexes Related to Small Grids

10.37236/6212 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Benjamin Braun ◽  
Wesley K. Hough
Keyword(s):  
Euler Characteristic ◽  
Grid Graph ◽  
Homology Groups ◽  
Matching Complex

The topology of the matching complex for the $2\times n$ grid graph is mysterious. We describe a discrete Morse matching for a family of independence complexes $\mathrm{Ind}(\Delta_n^m)$ that include these matching complexes. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups for certain $\mathrm{Ind}(\Delta_n^m)$. Further, we determine the Euler characteristic of $\mathrm{Ind}(\Delta_n^m)$ and prove that several homology groups of $\mathrm{Ind}(\Delta_n^m)$ are non-zero.

A CATEGORIFICATION FOR THE PENROSE POLYNOMIAL

2011 ◽  
Vol 20 (01) ◽  
pp. 141-157
Author(s):  
KERRY LUSE ◽  
YONGWU RONG
Keyword(s):  
Euler Characteristic ◽  
Jones Polynomial ◽  
Homology Groups ◽  
Tutte Polynomials

Given a graph, we construct homology groups whose Euler characteristic is the Penrose polynomial of the graph, evaluated at an integer. This work is motivated by Khovanov's work on the categorification of the Jones polynomial for knots, and the subsequent categorifications of the chromatic and Tutte polynomials for graphs.

scholarly journals On the Loop Homology of a Certain Complex of RNA Structures

2021 ◽  
Author(s):  
Thomas J.X. Li ◽  
Christian M. Reidys
Keyword(s):  
Euler Characteristic ◽  
Homology Group ◽  
Secondary Structures ◽  
Rna Structures ◽  
Sequence Structure ◽  
Evolutionary Transitions ◽  
Partial Matching ◽  
Rna Sequence ◽  
Homology Groups ◽  
Topological Framework

In this paper we establish a topological framework of τ-structures to quantify the evolutionary transitions between two RNA sequence-structure pairs. τ-structures developed here consist of a pair of RNA secondary structures together with a non-crossing partial matching between the two backbones. The loop complex of a τ-structure captures the intersections of loops in both secondary structures. We compute the loop homology of τ-structures. We show that only the zeroth, first and second homology groups are free. In particular, we prove that the rank of the second homology group equals the number γ of certain arc-components in a τ-structure, and the rank of the first homology is given by γ−χ+1, where χ is the Euler characteristic of the loop complex.

scholarly journals Profinite rigidity for twisted Alexander polynomials

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jun Ueki
Keyword(s):  
Rigidity Theorem ◽  
Homology Groups ◽  
Alexander Polynomials ◽  
Twisted Homology ◽  
Hyperbolic Volumes

AbstractWe formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a {{\mathbb{Z}}}-cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley’s parabolic representations of two-bridge knot groups and give a remark on hyperbolic volumes.

scholarly journals Vortices over Riemann surfaces and dominated splittings

2021 ◽  
pp. 1-26
Author(s):  
THOMAS METTLER ◽  
GABRIEL P. PATERNAIN
Keyword(s):  
Euler Characteristic ◽  
Tangent Bundle ◽  
Riemann Surfaces ◽  
Dominated Splitting ◽  
Vortex Equations ◽  
Special Cases ◽  
Unit Tangent Bundle ◽  
Anosov Flows

Abstract We associate a flow $\phi $ with a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi $ always admits a dominated splitting and identify special cases in which $\phi $ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$ .

scholarly journals Path homology theory of edge-colored graphs

2021 ◽  
Vol 19 (1) ◽  
pp. 706-723
Author(s):  
Yuri V. Muranov ◽  
Anna Szczepkowska
Keyword(s):  
Spectral Sequence ◽  
Edge Coloring ◽  
Homology Theory ◽  
Homotopy Category ◽  
Homology Groups ◽  
Colored Graphs ◽  
Natural Filtration ◽  
Definition Of ◽  
Exact Sequences

Abstract In this paper, we introduce the category and the homotopy category of edge-colored digraphs and construct the functorial homology theory on the foundation of the path homology theory provided by Grigoryan, Muranov, and Shing-Tung Yau. We give the construction of the path homology theory for edge-colored graphs that follows immediately from the consideration of natural functor from the category of graphs to the subcategory of symmetrical digraphs. We describe the natural filtration of path homology groups of any digraph equipped with edge coloring, provide the definition of the corresponding spectral sequence, and obtain commutative diagrams and braids of exact sequences.

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