Double Witt groups

TL;DR

This paper introduces the double Witt group, an algebraic refinement distinguishing slice and doubly-slice knots, and explores its properties and relationships with classical Witt groups and linking forms.

Contribution

It defines the double Witt group for linking forms, computes it for Dedekind domains, and establishes its isomorphism with the double Witt group of Seifert forms.

Findings

Calculated the double Witt group for Dedekind domains.

Established the relationship between double Witt and classical Witt groups.

Proved the isomorphism between double Witt groups of Seifert and Blanchfield forms.

Abstract

The difference between slice and doubly-slice knots is reflected in algebra by the difference between metabolic and hyperbolic Blanchfield linking forms. We exploit this algebraic distinction to refine the classical Witt group of linking forms by defining a `double Witt group' of linking forms. We calculate the double Witt group for Dedekind domains and precisely determine its relationship to the classical Witt group. Finally, we prove that the double Witt group of Seifert forms is isomorphic to the double Witt group of Blanchfield forms.