scholarly journals TORSION OF QUASI-ISOMORPHISMS

2009 ◽  
Vol 18 (09) ◽  
pp. 1227-1258
Author(s):  
JAE-WOOK CHUNG ◽  
XIAO-SONG LIN
Keyword(s):  
Mild Condition ◽  
Chain Complex ◽  
Reidemeister Torsion ◽  
Homotopy Invariant ◽  
Chain Complexes ◽  
Chain Homotopy ◽  
Chain Map ◽  
A Chain

In this paper, we introduce the notion of Reidemeister torsion for quasi-isomorphisms of based chain complexes over a field. We call a chain map a quasi-isomorphism if its induced homomorphism between homology is an isomorphism. Our notion of torsion generalizes the torsion of acyclic based chain complexes, and is a chain homotopy invariant on the collection of all quasi-isomorphisms from a based chain complex to another. It shares nice properties with torsion of acyclic based chain complexes, like multiplicativity and duality. We will further generalize our torsion to quasi-isomorphisms between free chain complexes over a ring under some mild condition. We anticipate that the study of torsion of quasi-isomorphisms will be fruitful in many directions, and in particular, in the study of links in 3-manifolds.

Projective Approximations

1984 ◽  
Vol 36 (1) ◽  
pp. 178-192 ◽  
Author(s):  
K. Varadarajan
Keyword(s):  
Associative Ring ◽  
Chain Complex ◽  
Chain Complexes ◽  
Chain Map ◽  
Chain Maps ◽  
A Chain

Let R be an associative ring with 1 ≠ 0. Throughout we will be considering unitary left R-modules. Given a chain complex C over R, a free approximation of C is defined to be a free chain complex F over R together with an epimorphism τ:F → C of chain complexes with the property that H(τ):H(F) ≃ H(C). In Chapter 5, Section 2 of [3] it is proved that any chain complex C over Z has a free approximation τ:F → C. Moreover given a free approximation τ:F → C of C and any chain map f:F’ → C with F’ a free chain complex over Z, there exists a chain map φ:F’→ F with T O φ = f . Any two chain maps φ, ψ of F’ in F with T O φ = T O ψ are chain homotopic.

scholarly journals AN INVARIANT OF LEGENDRIAN AND TRANSVERSE LINKS FROM OPEN BOOK DECOMPOSITIONS OF CONTACT 3-MANIFOLDS

2020 ◽  
pp. 1-33
Author(s):  
ALBERTO CAVALLO
Keyword(s):  
Chain Complex ◽  
Open Book ◽  
Rational Homology ◽  
Open Book Decompositions ◽  
Chain Complexes ◽  
Legendrian Links ◽  
Sum Formula ◽  
A Chain ◽  
Special Cycle ◽  
Rational Homology Sphere

Abstract We introduce a generalization of the Lisca–Ozsváth–Stipsicz–Szabó Legendrian invariant ${\mathfrak L}$ to links in every rational homology sphere, using the collapsed version of link Floer homology. We represent a Legendrian link L in a contact 3-manifold ${(M,\xi)}$ with a diagram D, given by an open book decomposition of ${(M,\xi)}$ adapted to L, and we construct a chain complex ${cCFL^-(D)}$ with a special cycle in it denoted by ${\mathfrak L(D)}$ . Then, given two diagrams ${D_1}$ and ${D_2}$ which represent Legendrian isotopic links, we prove that there is a map between the corresponding chain complexes that induces an isomorphism in homology and sends ${\mathfrak L(D_1)}$ into ${\mathfrak L(D_2)}$ . Moreover, a connected sum formula is also proved and we use it to give some applications about non-loose Legendrian links; that are links such that the restriction of ${\xi}$ on their complement is tight.

Relationships between the activity of respiratory-chain complexes in pre- (biopsy) or post-slaughter muscle samples and feed efficiency in random-bred Ghezel lambs

2017 ◽  
Vol 57 (8) ◽  
pp. 1674
Author(s):  
M. J. Zamiri ◽  
R. Mehrabi ◽  
G. R. Kavoosi ◽  
H. Rajaei Sharifabadi
Keyword(s):  
Respiratory Chain ◽  
Feed Intake ◽  
Enzyme Activities ◽  
Feed Efficiency ◽  
Average Daily Gain ◽  
Chain Complex ◽  
Respiratory Chain Complex ◽  
Respiratory Chain Complexes ◽  
Chain Complexes ◽  
Daily Gain

The present study was conducted to determine the relationship between the activity of mitochondrial respiratory chain complexes in pre- and post-slaughter muscle samples and residual feed intake (RFI) in Ghezel male lambs born as a result of random mating. The study was based on the hypothesis that random-bred lambs with lower feed (or higher) RFI have lower (or higher) respiratory chain-complex activity in muscle samples. Lambs (n = 30) were fed a diet consisting of 70% concentrate and 30% alfalfa hay during a 70-day period. Individual feed intake and average daily gain were recorded to calculate the RFI, feed-conversion ratio (FCR) and adjusted FCR (aFCR). On the basis of these calculations, the lambs were classified into low and high groups for RFI, with FCR and aFCR (n = 22) being one standard deviation above or below the means; this was corroborated by Student’s t-test (P < 0.01). At the end of the experiment, a 10-g biopsy sample was taken from the posterior side of the left femoral biceps. After 24 h, the lambs were slaughtered, and a sample from the posterior side of the right femoral biceps was dissected for determination of mitochondrial protein and respiratory chain-complex activities (Complexes I–V). The RFI was not correlated with the metabolic bodyweight and average daily gain, but was positively correlated (r = 0.56) with the average daily feed intake (P < 0.01); mean daily feed intake in the low-RFI group was 200 g less than that in the high-RFI group. The FCR and aFCR were not significantly (P > 0.05) correlated with average daily feed intake (r = 0.39 and r = 0.36 respectively), but showed a negative correlation (P < 0.01) with average daily gain (r = –0.73 and r = –0.76 respectively). Although very high negative correlations were recorded between the activities of all five respiratory-chain complexes and RFI in muscle samples obtained before (–0.91 to –0.97) and after (–0.92 to –0.97) slaughter, Complexes I and V showed small negative correlations (–0.40) with FCR or aFCR (P < 0.05). Enzyme activities of the respiratory-chain Complexes I, III and V were not significantly different between the pre- and post-slaughter biopsy samples; however, the enzyme activities of respiratory-chain Complexes II and IV were slightly higher in post-slaughter samples (P < 0.01). These results suggested that it may be possible to use the enzymatic activity of respiratory-chain complexes in muscle biopsy samples for screening of lambs for RFI, providing a useful procedure for genetic selection of lambs for this component of feed efficiency. These encouraging results need to be verified in further experiments using other sheep breeds and a larger number of lambs.

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 Synthesis, Crystal Structure, and N2-Adsorption Property of a Chain Complex of Rhodium(II) Benzoate and Piperazine

2013 ◽  
Vol 29 (0) ◽  
pp. 21-22 ◽  
Author(s):  
Masahiro MIKURIYA ◽  
Kazuya OUCHI ◽  
Yasutaka NAKANISHI ◽  
Daisuke YOSHIOKA ◽  
Hidekazu TANAKA ◽  
...  
Keyword(s):  
Crystal Structure ◽  
Adsorption Property ◽  
Chain Complex ◽  
N2 Adsorption ◽  
A Chain

scholarly journals EXPLICIT cocycle formulas on finite abelian groups with applications to braided linear Gr-categories and Dijkgraaf–Witten invariants

2019 ◽  
Vol 150 (4) ◽  
pp. 1937-1964 ◽  
Author(s):  
Hua-Lin Huang ◽  
Zheyan Wan ◽  
Yu Ye
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Finite Abelian Groups ◽  
Bar Resolution ◽  
Chain Map ◽  
A Chain

AbstractWe provide explicit and unified formulas for the cocycles of all degrees on the normalized bar resolutions of finite abelian groups. This is achieved by constructing a chain map from the normalized bar resolution to a Koszul-like resolution for any given finite abelian group. With a help of the obtained cocycle formulas, we determine all the braided linear Gr-categories and compute the Dijkgraaf–Witten Invariants of the n-torus for all n.

scholarly journals A VOLUME FORM ON THE KHOVANOV INVARIANT

2012 ◽  
Vol 21 (04) ◽  
pp. 1250032
Author(s):  
JUAN ORTIZ-NAVARRO
Keyword(s):  
Chain Complex ◽  
Khovanov Homology ◽  
Volume Form ◽  
Reidemeister Torsion ◽  
Homology Groups ◽  
Numerical Invariant ◽  
Reidemeister Moves ◽  
Invariant Volume ◽  
Knots And Links

The Reidemeister torsion construction can be applied to the chain complex used to compute the Khovanov homology of a knot or a link. This defines a volume form on Khovanov homology. The volume form transforms correctly under Reidemeister moves to give an invariant volume on the Khovanov homology. In this paper, its construction and invariance under these moves is demonstrated. Also, some examples of the invariant are presented for particular choices for the bases of homology groups to obtain a numerical invariant of knots and links. In these examples, the algebraic torsion seen in the Khovanov chain complex when homology is computed over ℤ is recovered.

scholarly journals CATEGORIFICATION OF THE DICHROMATIC POLYNOMIAL FOR GRAPHS

2008 ◽  
Vol 17 (01) ◽  
pp. 31-45 ◽  
Author(s):  
MARKO STOŠIĆ
Keyword(s):  
Positive Integer ◽  
Euler Characteristic ◽  
Jones Polynomial ◽  
Chain Complex ◽  
Tutte Polynomial ◽  
A Chain ◽  
Alternating Link ◽  
Dichromatic Polynomial

For each graph and each positive integer n, we define a chain complex whose graded Euler characteristic is equal to an appropriate n-specialization of the dichromatic polynomial. This also gives a categorification of n-specializations of the Tutte polynomial of graphs. Also, for each graph and integer n ≤ 2, we define the different one-variable n-specializations of the dichromatic polynomial, and for each polynomial, we define graded chain complex whose graded Euler characteristic is equal to that polynomial. Furthermore, we explicitly categorify the specialization of the Tutte polynomial for graphs which corresponds to the Jones polynomial of the appropriate alternating link.

scholarly journals FINITE DOMINATION AND NOVIKOV RINGS. ITERATIVE APPROACH

2012 ◽  
Vol 55 (1) ◽  
pp. 145-160 ◽  
Author(s):  
THOMAS HÜTTEMANN ◽  
DAVID QUINN
Keyword(s):  
Laurent Series ◽  
Chain Complex ◽  
Laurent Polynomial ◽  
Projective Modules ◽  
Finitely Generated ◽  
Formal Laurent Series ◽  
Chain Complexes ◽  
Laurent Polynomial Ring ◽  
Bounded Below ◽  
Free Modules

AbstractSuppose C is a bounded chain complex of finitely generated free modules over the Laurent polynomial ring L = R[x,x−1]. Then C is R-finitely dominated, i.e. homotopy equivalent over R to a bounded chain complex of finitely generated projective R-modules if and only if the two chain complexes C ⊗LR((x)) and C ⊗LR((x−1)) are acyclic, as has been proved by Ranicki (A. Ranicki, Finite domination and Novikov rings, Topology34(3) (1995), 619–632). Here R((x)) = R[[x]][x−1] and R((x−1)) = R[[x−1]][x] are rings of the formal Laurent series, also known as Novikov rings. In this paper, we prove a generalisation of this criterion which allows us to detect finite domination of bounded below chain complexes of projective modules over Laurent rings in several indeterminates.

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