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

Abstract

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

$$\sum _{I\subset N}{\left(-1\right)}^{♯I}{Z}_n\left(M_I\right)$$

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 $2n$ 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 $Z_n$ 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 $\mathbb{Q}∕\mathbb{Z}$ is fixed, and the coefficients of these polynomials are knot invariants, for which various topological properties or topological definitions are given.