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The Structure of the Rational Concordance Group of Knots

Jae Choon Cha Information and Communications University, Daejeon, Korea
Available Formats:
Electronic 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 Click above image for expanded view The Structure of the Rational Concordance Group of Knots Jae Choon Cha Information and Communications University, Daejeon, Korea Available Formats:  Electronic 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.

• 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 reviewers who would like to review an AMS book
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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 reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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