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

Abstract

We first present three 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 three formulae determines an alternate sum of the form $$\sum _{I\subset N}(-1{)}^{♯I}{Z}_n(M_I)$$ where $N$ is the set of components of a framed algebraically split link $L$ in a rational homology sphere $M$, and $M_I$ denotes the manifold resulting from the Dehn surgeries on the components of $I$. The first formula treats the case when $L$ is a boundary link with $n$ components, while the second one is for $3n$--component algebraically split links. In the third formula, the link $L$ has $2n$ components and the Milnor triple linking numbers of its 3--component sublinks vanish. 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 $\QQ/\ZZ$ is fixed, and the coefficients of these polynomials are knot invariants, for which various topological properties or topological definitions are given.

51 pages, uses pstricks

References in corpus (3)

• Splitting formulae for the Kontsevich-Kuperberg-Thurston invariant of rational homology 3-spheres
• On the Kontsevich-Kuperberg-Thurston construction of a configuration-space invariant for rational homology 3-spheres
• Clover calculus for homology 3-spheres via basic algebraic topology

Cited by in corpus (7)

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• Alternating odd pretzel knots and chirally cosmetic surgeries
• A note on Jones polynomial and cosmetic surgery
• An invariant of rational homology 3-spheres via vector fields
• On elementary invariants of genus one knots and Seifert surfaces