SMF

Un algorithme pour calculer l'invariant de Walker

Christine Lescop
Un algorithme pour calculer l'invariant de Walker
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  • Année : 1990
  • Fascicule : 3
  • Tome : 118
  • Format : Électronique
  • Langue de l'ouvrage :
    Français
  • Pages : 363-376
  • DOI : 10.24033/bsmf.2150
En 1988, Kevin Walker a étendu l'invariant de Casson aux sphères d'homologie rationnelle. Cette note décrit un procédé simple et programmable pour calculer l'invariant de Walker d'une sphère d'homologie rationnelle connue par l'un de ses diagrammes de chirurgies. (On appelle diagramme de chirurgie la projection régulière d'un entrelacs dont chaque composante est pondérée par un rationnel.)
In his thesis, Kevin Walker extends the invariant defined by A. Casson for homology $3$-spheres to rational homology spheres ($3$-manifolds with the same rational homology as $S^3$). The modification of Walker's invariant after a surgery on a single knot which transforms a rational homology sphere to another rational homology sphere is given by a Surgery Formula (see paragraph 0). In this paper, we describe a simple and programmable process to calculate Walker's invariant of a rational homology sphere given by one of its surgery diagrams. (We call a surgery diagram a regular projection of a link, each component of which is framed by a rational number ; we use Rolfsen's conventions to define the manifold presented by such a diagram (see [R], p. 259–260)). The first section shows how to modify the diagram in a simple way so that the computation reduces to a finite sequence of applications of the Surgery Formula. The second section is devoted to describe elementary methods to do it. Section 0 states a few results of Walker.