Homology Invariants

1978 ◽  
Vol 30 (03) ◽  
pp. 655-670 ◽  
Author(s):  
Richard Hartley ◽  
Kunio Murasugi
Keyword(s):  
Homology Group ◽  
Covering Space ◽  
Simple Relationship ◽  
Cyclic Covering ◽  
Covering Spaces ◽  
Homology Groups ◽  
Branched Covering ◽  
The Relationship

There have been few published results concerning the relationship between the homology groups of branched and unbranched covering spaces of knots, despite the fact that these invariants are such powerful invariants for distinguishing knot types and have long been recognised as such [8]. It is well known that a simple relationship exists between these homology groups for cyclic covering spaces (see Example 3 in § 3), however for more complicated covering spaces, little has previously been known about the homology group, H1(M) of the branched covering space or about H1(U), U being the corresponding unbranched covering space, or about the relationship between these two groups.

Strongly-cyclic branched coverings and the Alexander polynomial of knots in rational homology spheres

2007 ◽  
Vol 142 (2) ◽  
pp. 259-268 ◽  
Author(s):  
YUYA KODA
Keyword(s):  
Fundamental Group ◽  
Homology Group ◽  
Alexander Polynomial ◽  
Homology Sphere ◽  
Cyclic Covering ◽  
Rational Homology ◽  
Branched Coverings ◽  
Branched Covering ◽  
Rational Homology Spheres ◽  
Rational Homology Sphere

AbstractLet K be a knot in a rational homology sphere M. In this paper we correlate the Alexander polynomial of K with a g-word cyclic presentation for the fundamental group of the strongly-cyclic covering of M branched over K. We also give a formula for the order of the first homology group of the strongly-cyclic branched covering.

Construction and visualization of branched covering spaces

2016 ◽  
Author(s):  
Sanaz Golbabaei ◽  
Lawrence Roy ◽  
Prashant Kumar ◽  
Eugene Zhang
Keyword(s):  
Covering Spaces ◽  
Branched Covering

scholarly journals One-relator groups and algebras related to polyhedral products

2021 ◽  
pp. 1-20
Author(s):  
Jelena Grbić ◽  
George Simmons ◽  
Marina Ilyasova ◽  
Taras Panov
Keyword(s):  
Homology Group ◽  
Homotopy Theory ◽  
Commutator Subgroup ◽  
Homology Groups ◽  
Homological Characterization ◽  
One Relator Groups ◽  
Combinatorial Condition ◽  
The Moment ◽  
Necessary And Sufficient

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$ , we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $\mathcal {R}_K$ , to be a one-relator group; and for the Pontryagin algebra $H_{*}(\Omega \mathcal {Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterization of these properties. For $RC_K'$ , it is given by a condition on the homology group $H_2(\mathcal {R}_K)$ , whereas for $H_{*}(\Omega \mathcal {Z}_K)$ it is stated in terms of the bigrading of the homology groups of $\mathcal {Z}_K$ .

scholarly journals The Decay of Disease Association with Declining Linkage Disequilibrium: A Fine Mapping Theorem

2016 ◽  
Author(s):  
Mehdi Maadooliat ◽  
Naveen K. Bansal ◽  
Jiblal Upadhya ◽  
Manzur R. Farazi ◽  
Zhan Ye ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Linkage Disequilibrium ◽  
Fine Mapping ◽  
Disease Model ◽  
Disease Association ◽  
Simple Relationship ◽  
Locus System ◽  
Using Data ◽  
Relationship Of ◽  
The Relationship

AbstractSeveral important and fundamental aspects of disease genetics models have yet to be described. One such property is the relationship of disease association statistics at a marker site closely linked to a disease causing site. A complete description of this two-locus system is of particular importance to experimental efforts to fine map association signals for complex diseases. Here, we present a simple relationship between disease association statistics and the decline of linkage disequilibrium from a causal site. A complete derivation of this relationship from a general disease model is shown for very large sample sizes. Quite interestingly, this relationship holds across all modes of inheritance. Extensive Monte Carlo simulations using a disease genetics model applied to chromosomes subjected to a standard model of recombination are employed to better understand the variation around this fine mapping theorem due to sampling effects. We also use this relationship to provide a framework for estimating properties of a non-interrogated causal site using data at closely linked markers. We anticipate that understanding the patterns of disease association decay with declining linkage disequilibrium from a causal site will enable more powerful fine mapping methods.

scholarly journals Studies on the antibacterial properties of the bile acids and some compounds derived from cholanic acid

1947 ◽  
Vol 134 (877) ◽  
pp. 523-537 ◽  
Keyword(s):  
Molecular Structure ◽  
Staphylococcus Aureus ◽  
Surface Tension ◽  
Bile Acids ◽  
Antibacterial Properties ◽  
Simple Relationship ◽  
The Relationship ◽  
Positive Complex ◽  
Cholanic Acid ◽  
Limiting Dilution

In this work the bile acids and certain of their derivatives have been studied with regard to their bacteriostatic power and relative activities in removing the Gram-positive complex from yeast. No correlation between these properties was obtained. A simple relationship was apparent, however, between the bacteriostatic activities of the compounds and their ability to depress the surface tension of the metabolism medium. The limiting dilution at which the 'active' bile acids were bacteriostatic for Staphylococcus aureus corresponded to a depression of the surface tension of the medium by approximately 4∙5 dynes. The relationship between bacteriostatic power and surface activity was only valid for this particular series of compounds of closely related molecular structure.

Minimal Plat Representations of Prime Knots and Links are not Unique

1976 ◽  
Vol 28 (1) ◽  
pp. 161-167 ◽  
Author(s):  
José M. Montesinos
Keyword(s):  
Infinite Family ◽  
Equivalence Classes ◽  
Covering Space ◽  
Heegaard Splittings ◽  
Cyclic Covering ◽  
Knots And Links

Letdenote the 2-fold cyclic covering space branched over a linkLin S3. We wish to describe an infinite family of prime knots and links in which each memberLexhibits two minimal 6-plat representations, where the associated Heegaard splittings ofare minimal and inequivalent. Thus each knot or link of that family admits at least two equivalence classes of 6-plat representations which are minimal.

scholarly journals Multivariate Alexander colorings

2018 ◽  
Vol 27 (14) ◽  
pp. 1850076 ◽  
Author(s):  
Lorenzo Traldi
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Knot Theory ◽  
Laurent Polynomial ◽  
Module Structure ◽  
Cyclic Covering ◽  
Covering Spaces ◽  
Linear Maps ◽  
Laurent Polynomial Ring ◽  
Special Case

We extend the notion of link colorings with values in an Alexander quandle to link colorings with values in a module [Formula: see text] over the Laurent polynomial ring [Formula: see text]. If [Formula: see text] is a diagram of a link [Formula: see text] with [Formula: see text] components, then the colorings of [Formula: see text] with values in [Formula: see text] form a [Formula: see text]-module [Formula: see text]. Extending a result of Inoue [Knot quandles and infinite cyclic covering spaces, Kodai Math. J. 33 (2010) 116–122], we show that [Formula: see text] is isomorphic to the module of [Formula: see text]-linear maps from the Alexander module of [Formula: see text] to [Formula: see text]. In particular, suppose [Formula: see text] is a field and [Formula: see text] is a homomorphism of rings with unity. Then [Formula: see text] defines a [Formula: see text]-module structure on [Formula: see text], which we denote [Formula: see text]. We show that the dimension of [Formula: see text] as a vector space over [Formula: see text] is determined by the images under [Formula: see text] of the elementary ideals of [Formula: see text]. This result applies in the special case of Fox tricolorings, which correspond to [Formula: see text] and [Formula: see text]. Examples show that even in this special case, the higher Alexander polynomials do not suffice to determine [Formula: see text]; this observation corrects erroneous statements of Inoue [Quandle homomorphisms of knot quandles to Alexander quandles, J. Knot Theory Ramifications 10 (2001) 813–821; op. cit.].

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