Rational Blanchfield forms, S-equivalence, and null LP-surgeries

Delphine Moussard (1)

1 IF - Institut Fourier

Garoufalidis and Rozansky introduced null-moves on the set of pairs (M,K), where M is an integral homology sphere and K is a knot in M. These null-moves are suitable to study the Kricker lift of the Kontsevich integral. They defined a filtration on the space generated by pairs (M,K) up to orientation-preserving homeomorphism. This filtration splits with respect to the isomorphism classes of integral Alexander modules equipped with their Blanchfield forms. Null Lagrangian-preserving surgeries are a generalization of the Garoufalidis and Rozansky null-moves 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\setminus K by other such handlebodies with identical Lagrangian. We prove that two pairs (M,K) can be obtained from one another by a finite sequence of null Lagrangian-preserving surgeries if and only if they have isomorphic rational Alexander modules and Blanchfield forms.

Mots clés: Alexander module; Blanchfield form; equivariant linking pairing; homology sphere; homology handlebody; Lagrangian-preserving surgery; Seifert matrix; S-equivalence; lift of the Kontsevich integral; null-move; Euler degree of the Kontsevich integral.; Euler degree of the Kontsevich integral

Domaines: Topologie algébrique [math.AT]; Topologie géométrique [math.GT]

Delphine Moussard. Rational Blanchfield forms, S-equivalence, and null LP-surgeries. 2012. ⟨hal-00715583v1⟩