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Surgery formulae for finite type invariants of rational homology $3$–spheres

Christine Lescop

Algebraic & Geometric Topology 9 (2009) 979–1047

We first present four graphic surgery formulae for the degree n part Zn of the Kontsevich–Kuperberg–Thurston universal finite type invariant of rational homology spheres.

Each of these four formulae determines an alternate sum of the form

IN(1)IZ n(MI)

where N is a finite set of disjoint operations to be performed on a rational homology sphere M, and MI denotes the manifold resulting from the operations in I. The first formula treats the case when N is a set of 2n Lagrangian-preserving surgeries (a Lagrangian-preserving surgery replaces a rational homology handlebody by another such without changing the linking numbers of curves in its exterior). In the second formula, N is a set of n Dehn surgeries on the components of a boundary link. The third formula deals with the case of 3n surgeries on the components of an algebraically split link. The fourth formula is for 2n surgeries on the components of an algebraically split link in which all Milnor triple linking numbers vanish. In the case of homology spheres, these formulae can be seen as a refinement of the Garoufalidis–Goussarov–Polyak comparison of different filtrations of the rational vector space freely generated by oriented homology spheres (up to orientation preserving homeomorphisms).

The presented formulae are then applied to the study of the variation of Zn under a pq–surgery on a knot K. This variation is a degree n polynomial in qp when the class of qp in is fixed, and the coefficients of these polynomials are knot invariants, for which various topological properties or topological definitions are given.

finite type invariant, 3-manifold, surgery formula, Jacobi diagram, Casson–Walker invariant, configuration space invariant, Goussarov–Habiro filtration, clasper, clover, Y-graph
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 57N10, 57M25, 55R80
Received: 1 April 2009
Accepted: 2 April 2009
Published: 16 April 2009
Christine Lescop
Institut Fourier
BP 74
38402 Saint-Martin d’Hères cedex