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.

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.

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 Matching Complexes of Small Grids

10.37236/8480 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Takahiro Matsushita
Keyword(s):  
Simplicial Complex ◽  
Line Graph ◽  
Simple Graph ◽  
Grid Graph ◽  
Matching Complex ◽  
Independence Complex

The matching complex $M(G)$ of a simple graph $G$ is the simplicial complex consisting of the matchings on $G$. The matching complex $M(G)$ is isomorphic to the independence complex of the line graph $L(G)$.  Braun and Hough introduced a family of graphs $\Delta^m_n$, which is a generalization of the line graph of the $(n \times 2)$-grid graph. In this paper, we show that the independence complex of $\Delta^m_n$ is a wedge of spheres. This gives an answer to a problem suggested by Braun and Hough.

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 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.

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