ETD

• concordance
• knots and links
• Milnor invariants
• quantum invariants
• Whitney towers

In Chapter 1 we explain the theory of Whitney towers in its simplest, framed form and show its connection with a class of link invariants defined by Milnor in his early works and with the problem of deciding whether a link is slice (i.e. concordant to unlink) or not. In Chapter 2 we address some subtler problem regarding twisting in Whitney towers and the corresponding obstruction theory. Twisted towers are used to define a notion of twisted concordance and a corresponding filtration on the set of links. The structure of this filtration is not fully understood yet and it seems to suggest the existence of a new class of concordance invariants generalizing the classical Arf invariant of knots. In Chapter 3 we explain a general procedure to build string link invariants from ribbon Hopf algebras and a recent result of Meilhan and Suzuki describing concordance information contained in the invariant associated to the h-adic quantized universal enveloping algebra of the Lie algebra sl_2. The hope is that quantum invariants coming from other ribbon Hopf algebras could also contain concordance information and be related to the obstruction theory of twisted Whitney towers.