  Null Lagrangian-preserving surgeries are a generalization of the Garoufalidis
and Rozansky null-moves, that these authors introduced to study the Kricker
lift of the Kontsevich integral, in the setting of pairs (M,K) composed of a
rational homology sphere M and a null-homologous knot K in M. They are defined
as replacements of null-homologous rational homology handlebodies of M\K by
other such handlebodies with identical Lagrangian. A null Lagrangian-preserving
surgery induces a canonical isomorphism between the rational Alexander modules
of the involved pairs, which preserves the Blanchfield form. Conversely, we
prove that a fixed isomorphism between rational Alexander modules which
preserves the Blanchfield form can be realized, up to multiplication by a power
of t, by a finite sequence of null Lagrangian-preserving surgeries. We also
prove that such classes of isomorphisms can be realized by rational
S-equivalences. In the case of integral homology spheres, we prove similar
realization results for a fixed isomorphism between integral Alexander modules.

Algebraic Topology;Geometric Topology