Lower Order Solvability, Seifert Forms, and Blanchfield Forms of Links
Doctor of Philosophy
We define and study specific generalizations of Seifert forms and Blanchfield forms to links and study their relationships with lower order solvability and with each other. We define Seifert Z-surfaces for links with pairwise linking numbers zero and prove that if a link is 0.5-solvable then every Seifert Z-surface has a metabolizer. We use this result to determine that Arf invariants and Milnor's invariants are not sufficient to classify 0.5-solvable links. We define nonsingular localized Blanchfield forms for links with pairwise linking numbers zero and build on work of Cochran-Orr-Teichner and Cochran-Harvey-Leidy to show that 1-solvability implies each of these Blanchfield forms are hyperbolic. We also define Blanchfield forms on the infinite cyclic covers of the exterior of a link with pairwise linking numbers zero and build on work of Friedl-Powell to prove that in a special case, a Seifert Z-surface having a metabolizer implies the Blanchfield form is hyperbolic. There are well known definitions of boundary Seifert surfaces and multivariable Blanchfield forms for boundary links. We define a boundary metabolizer for a boundary Seifert surface, which is more restrictive than the usual definition of a metabolizer, and prove that the existence of a boundary metabolizer implies both 0.5-solvability and that the multivariable Blanchfield form is hyperbolic.