scholarly journals A characterisation of the $\mathbf{n\langle1\rangle \oplus\langle3\rangle}$ form and applications to rational homology spheres

2006 ◽  
Vol 13 (2) ◽  
pp. 259-271 ◽  
Author(s):  
Brendan Owens ◽  
Saso Strle
Keyword(s):  
Rational Homology ◽  
Rational Homology Spheres

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.

scholarly journals ON ALEXANDER MODULES AND BLANCHFIELD FORMS OF NULL-HOMOLOGOUS KNOTS IN RATIONAL HOMOLOGY SPHERES

2012 ◽  
Vol 21 (05) ◽  
pp. 1250042 ◽  
Author(s):  
DELPHINE MOUSSARD
Keyword(s):  
Isomorphism Class ◽  
The Other ◽  
Rational Homology ◽  
Rational Homology Spheres

In this paper, we give a classification of Alexander modules of null-homologous knots in rational homology spheres. We characterize these modules [Formula: see text] equipped with their Blanchfield forms ϕ, and the modules [Formula: see text] such that there is a unique isomorphism class of [Formula: see text], and we prove that for the other modules [Formula: see text], there are infinitely many such classes. We realize all these [Formula: see text] by explicit knots in ℚ-spheres.

scholarly journals LINEAR INDEPENDENCE IN THE RATIONAL HOMOLOGY COBORDISM GROUP

2019 ◽  
pp. 1-12 ◽  
Author(s):  
Marco Golla ◽  
Kyle Larson
Keyword(s):  
Infinite Order ◽  
Double Cover ◽  
Rational Homology ◽  
Cobordism Group ◽  
Connected Sums ◽  
Knot Concordance ◽  
Linearly Independent ◽  
Rational Homology Spheres ◽  
Homology Cobordism ◽  
Correction Terms

We give simple homological conditions for a rational homology 3-sphere $Y$ to have infinite order in the rational homology cobordism group $\unicode[STIX]{x1D6E9}_{\mathbb{Q}}^{3}$ , and for a collection of rational homology spheres to be linearly independent. These translate immediately to statements about knot concordance when $Y$ is the branched double cover of a knot, recovering some results of Livingston and Naik. The statements depend only on the homology groups of the 3-manifolds, but are proven through an analysis of correction terms and their behavior under connected sums.

scholarly journals Einstein metrics on rational homology spheres

2006 ◽  
Vol 74 (3) ◽  
pp. 353-362 ◽  
Author(s):  
Charles P. Boyer ◽  
Krzysztof Galicki
Keyword(s):  
Einstein Metrics ◽  
Rational Homology ◽  
Rational Homology Spheres

A COMBINATORIAL APPROACH TO LINKING NUMBERS IN RATIONAL HOMOLOGY SPHERES

2000 ◽  
Vol 09 (05) ◽  
pp. 703-711
Author(s):  
CARL J. STITZ
Keyword(s):  
Combinatorial Approach ◽  
Rational Homology ◽  
Characteristic Curves ◽  
Combinatorial Description ◽  
Linking Number ◽  
Reidemeister Moves ◽  
Linking Numbers ◽  
Heegaard Diagram ◽  
Rational Homology Spheres ◽  
Matrix Invariants

In this paper we find a method to compute the classical Seifert-Threlfall linking number for rational homology spheres without using 2-chains bounded by the curves in question. By using a Heegaard diagram for the manifold, we describe link isotopy combinatorially using the three traditional Reidemeister moves along with a fourth move which is essentially a Kirby move along the characteristic curves. This result is mathematical folklore which we set in print. We then use this combinatorial description of link isotopy to develop and prove the invariance of linking numbers. Once the linking numbers are in place, matrix invariants such as the Alexander polynomial can be computed.

scholarly journals On the Reidemeister torsion of rational homology spheres

2001 ◽  
Vol 25 (1) ◽  
pp. 11-17 ◽  
Author(s):  
Liviu I. Nicolaescu
Keyword(s):  
Quadratic Function ◽  
Reidemeister Torsion ◽  
Rational Homology ◽  
Rational Homology Spheres

We prove that theModℤreduction of the Reidemeister torsion of a rational homology 3-sphere is naturally aℚ/ℤ-valued quadratic function uniquely determined by aℚ/ℤ-constant and the linking form.

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