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The Structure of the Rational Concordance Group of Knots
 
Jae Choon Cha Information and Communications University, Daejeon, Korea
The Structure of the Rational Concordance Group of Knots
eBook ISBN:  978-1-4704-0489-5
Product Code:  MEMO/189/885.E
List Price: $66.00
MAA Member Price: $59.40
AMS Member Price: $39.60
The Structure of the Rational Concordance Group of Knots
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The Structure of the Rational Concordance Group of Knots
Jae Choon Cha Information and Communications University, Daejeon, Korea
eBook ISBN:  978-1-4704-0489-5
Product Code:  MEMO/189/885.E
List Price: $66.00
MAA Member Price: $59.40
AMS Member Price: $39.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1892007; 95 pp
    MSC: Primary 57;

    The author studies the group of rational concordance classes of codimension two knots in rational homology spheres. He gives a full calculation of its algebraic theory by developing a complete set of new invariants. For computation, he relates these invariants with limiting behaviour of the Artin reciprocity over an infinite tower of number fields and analyzes it using tools from algebraic number theory. In higher dimensions it classifies the rational concordance group of knots whose ambient space satisfies a certain cobordism theoretic condition. In particular, he constructs infinitely many torsion elements. He shows that the structure of the rational concordance group is much more complicated than the integral concordance group from a topological viewpoint. He also investigates the structure peculiar to knots in rational homology 3-spheres. To obtain further nontrivial obstructions in this dimension, he develops a technique of controlling a certain limit of the von Neumann \(L^2\)-signature invariants.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Rational knots and Seifert matrices
    • 3. Algebraic structure of $\mathcal {G}_n$
    • 4. Geometric structure of $\mathcal {C}_n$
    • 5. Rational knots in dimension three
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1892007; 95 pp
MSC: Primary 57;

The author studies the group of rational concordance classes of codimension two knots in rational homology spheres. He gives a full calculation of its algebraic theory by developing a complete set of new invariants. For computation, he relates these invariants with limiting behaviour of the Artin reciprocity over an infinite tower of number fields and analyzes it using tools from algebraic number theory. In higher dimensions it classifies the rational concordance group of knots whose ambient space satisfies a certain cobordism theoretic condition. In particular, he constructs infinitely many torsion elements. He shows that the structure of the rational concordance group is much more complicated than the integral concordance group from a topological viewpoint. He also investigates the structure peculiar to knots in rational homology 3-spheres. To obtain further nontrivial obstructions in this dimension, he develops a technique of controlling a certain limit of the von Neumann \(L^2\)-signature invariants.

  • Chapters
  • 1. Introduction
  • 2. Rational knots and Seifert matrices
  • 3. Algebraic structure of $\mathcal {G}_n$
  • 4. Geometric structure of $\mathcal {C}_n$
  • 5. Rational knots in dimension three
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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