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**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’.

Reviewer: J.Vrabec (Ljubljana)

### MSC:

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

57N12 | Topology of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010) |

57R65 | Surgery and handlebodies |

### Keywords:

Casson’s invariant; integral homology 3-spheres; rational homology 3- sphere; Dehn surgery; algorithm; surgery diagram### Citations:

Zbl 0699.57008
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\textit{C. Lescop}, Bull. Soc. Math. Fr. 118, No. 3, 363--376 (1990; Zbl 0718.57007)

### References:

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[3] | MARIN (A.) . - Un nouvel invariant pour les sphères d’homologie de dimension 3 , (d’après Casson), Séminaire Bourbaki, n^\circ 693, Février 1988 . Numdam | Zbl 0674.57013 · Zbl 0674.57013 |

[4] | ROLFSEN (D.) . - Knots and links . - Publish or Perish, 1976 . MR 58 #24236 | Zbl 0339.55004 · Zbl 0339.55004 |

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[6] | WALKER (K.) . - An extension of Casson’s Invariant , Thesis, University of California at Berkeley, in preparation, preliminary draft November 8, 1989 . |

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