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

We first present four graphic surgery formulae for the degree $n$ part $Z_{n}$ 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

\sum_{I \subset N} {(- 1)}^{♯ I} Z_{n} (M_{I})

where $N$ is a finite set of disjoint operations to be performed on a rational homology sphere $M$ , and $M_{I}$ denotes the manifold resulting from the operations in $I$ . The first formula treats the case when $N$ is a set of $2 n$ 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 $3 n$ surgeries on the components of an algebraically split link. The fourth formula is for $2 n$ 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 $Z_{n}$ under a $p ∕ q$ –surgery on a knot $K$ . This variation is a degree $n$ polynomial in $q ∕ p$ when the class of $q ∕ p$ in $ℚ ∕ ℤ$ is fixed, and the coefficients of these polynomials are knot invariants, for which various topological properties or topological definitions are given.

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Keywords

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

References

Publication

Received: 1 April 2009

Accepted: 2 April 2009

Published: 16 April 2009

Authors

Christine Lescop
Institut Fourier BP 74 38402 Saint-Martin d’Hères cedex France
http://www-fourier.ujf-grenoble.fr/~lescop/

Volume 9, issue 2 (2009)

Christine Lescop

Abstract