## Un algorithme pour calculer l’invariant de Walker. (An algorithm for the computation of Walker’s invariant).(French)Zbl 0718.57007

K. Walker [Bull. Am. Math. Soc., New Ser. 22, 261-267 (1990; Zbl 0699.57008)] has extended A. Casson’s $${\mathbb{Z}}$$-valued invariant, defined for oriented integral homology 3-spheres, to a $${\mathbb{Q}}$$-valued invariant $$\lambda$$ (M) defined for an arbitrary oriented rational homology 3- sphere (RHS) M. An important part of his theory is his surgery formula which, for RHS’s M and N such that N is obtained from M by a Dehn surgery along a knot in M, expresses $$\lambda$$ (N)-$$\lambda$$ (M) in terms of the surgery data. In the present paper the author develops a programmable algorithm for calculating $$\lambda$$ (M) for an RHS M which is given in terms of a surgery diagram, i.e. a link diagram D with a rational ‘weight’ assigned to each of its components such that the Dehn surgery on $$S^ 3$$ along D using the ‘weights’ as surgery coefficients (or ‘framing numbers’) produces M. She first shows how to replace D by another surgery diagram, $$D'$$, for M and how to order the components of $$D'$$ so that if one does the Dehn surgeries along the components of $$D'$$ (in the chosen order) one at a time, all intermediate manifolds are RHS’s. Then she describes how one can calculate the surgery data (which appear in Walker’s surgery formula) of all these one-component surgeries in terms of $$D'$$ and its ‘weights’.

### MSC:

 57N10 Topology of general $$3$$-manifolds (MSC2010) 57N12 Topology of the Euclidean $$3$$-space and the $$3$$-sphere (MSC2010) 57R65 Surgery and handlebodies

Zbl 0699.57008
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### References:

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