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Advances in the Mathematical Sciences
Differential and Symplectic Topology of Knots and Curves
Edited by: S. Tabachnikov, University of Arkansas at Fayetteville, AR

American Mathematical Society Translations--Series 2
Advances in the Mathematical Sciences
1999; 286 pp; hardcover
Volume: 190
ISBN-10: 0-8218-1354-4
ISBN-13: 978-0-8218-1354-6
List Price: US$120
Member Price: US$96
Order Code: TRANS2/190
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This book presents a collection of papers on two related topics: topology of knots and knot-like objects (such as curves on surfaces) and topology of Legendrian knots and links in 3-dimensional contact manifolds.

Featured is the work of international experts in knot theory ("quantum" knot invariants, knot invariants of finite type), in symplectic and contact topology, and in singularity theory. The interplay of diverse methods from these fields makes this volume unique in the study of Legendrian knots and knot-like objects such as wave fronts. A particularly enticing feature of the volume is its international significance. The volume successfully embodies a fine collaborative effort by worldwide experts from Belgium, France, Germany, Israel, Japan, Poland, Russia, Sweden, the U.K., and the U.S.


Graduate students and research mathematicians working in manifolds and cell complexes.

Table of Contents

  • J. C. Álvarez Paiva -- Contact topology, taut immersions, and Hilbert's fourth problem
  • E. Ferrand -- On Legendre cobordisms
  • V. Goryunov -- Vassiliev invariants of knots in \(\mathbb R^3\) and in a solid torus
  • T. Januszkiewicz and J. Świątkowski -- Finite type invariants of generic immersions of \(M^n\) into \(\mathbb R^{2n}\) are trivial
  • S. K. Lando -- On enumeration of unicursal curves
  • A. B. Merkov -- Vassiliev invariants classify flat braids
  • M. Polyak -- New Whitney-type formulas for plane curves
  • B. Shapiro -- Tree-like curves and their number of inflection points
  • S. Tabachnikov -- Geometry of exact transverse line fields and projective billiards
  • V. Tchernov -- Shadows of wave fronts and Arnold-Bennequin type invariants of fronts on surfaces and orbifolds
  • M. Umehara -- A unified approach to the four vertex theorems. I
  • G. Thorbergsson and M. Umehara -- A unified approach to the four vertex theorems. II
  • V. A. Vassiliev -- Topology of two-connected graphs and homology of spaces of knots
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