Higher-order linking forms
Constance Leidy · Rice Research Repository (Rice University) · 2004
Trotter [T] found examples of knots that have isomorphic classical Alexander modules, but non-isomorphic classical Blanchfield linking forms. T. Cochran [C] defined higher-order Alexander modules, An , (K), of a knot, K, and higher-order linking forms, Bℓn (K), which are linking forms defined on An , (K). When n = 0, these invariants are just the classical Alexander module and Blanchfield linking form. The question was posed in [C] whether Trotter's result generalized to the higher-order invariants. We show that it does. That is, we construct examples of knots that have isomorphic nth-order Alexander modules, but non-isomorphic nth-order linking forms. Furthermore, we define new higher-order linking forms on the Alexander modules for 3-manifolds considered by S. Harvey [H]. We construct examples of 3-manifolds with isomorphic nth-order Alexander modules, but non-isomorphic nth-order linking forms.