We first present four graphic surgery formulae for the degree
part
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
where
is
a finite set of disjoint operations to be performed on a rational homology sphere
, and
denotes the manifold resulting from the operations in
. The first formula
treats the case when
is a set of
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,
is a
set of
Dehn surgeries on the components of a boundary link. The third formula deals with the
case of
surgeries on the components of an algebraically split link. The fourth formula is for
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
under a
–surgery on a knot
. This variation
is a degree
polynomial in
when the class of
in
is
fixed, and the coefficients of these polynomials are knot invariants, for which various
topological properties or topological definitions are given.
Keywords
finite type invariant, 3-manifold, surgery formula, Jacobi
diagram, Casson–Walker invariant, configuration space
invariant, Goussarov–Habiro filtration, clasper, clover,
Y-graph
