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The Structure of the Rational Concordance Group of Knots
Jae Choon Cha, Information and Communications University, Daejeon, Korea

Memoirs of the American Mathematical Society
2007; 95 pp; softcover
Volume: 189
ISBN-10: 0-8218-3993-4
ISBN-13: 978-0-8218-3993-5
List Price: US$66
Individual Members: US$39.60
Institutional Members: US$52.80
Order Code: MEMO/189/885
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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

  • Introduction
  • Rational knots and Seifert matrices
  • Algebraic structure of \(\mathcal{G}_n\)
  • Geometric structure of \(\mathcal{C}_n\)
  • Rational knots in dimension three
  • Bibliography
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