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

TL;DR

This paper explores the relationship between null Lagrangian-preserving surgeries, rational Alexander modules, and Blanchfield forms, establishing how isomorphisms preserving these forms can be realized through specific surgeries and S-equivalences.

Contribution

It demonstrates that isomorphisms preserving the Blanchfield form can be realized via null Lagrangian-preserving surgeries and rational S-equivalences, extending understanding of knot invariants in rational homology spheres.

Findings

Null surgeries induce canonical isomorphisms preserving Blanchfield forms.

Such isomorphisms can be realized by finite sequences of surgeries.

Results extend to integral Alexander modules in homology spheres.

Abstract

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.