  This thesis consists of three self-contained chapters. The first two concern
quantum invariants of links and three manifolds and the third contains results
on the word problem for link groups.
  In chapter 1 we relate the tree part of the Aarhus integral to the
mu-invariants of string-links in homology balls thus generalizing results of
Habegger and Masbaum.
  There is a folklore result in physics saying that the Feynman integration of
an exponential is itself an exponential. In chapter 2 we state and prove an
exact formulation of this statement in the language which is used in the theory
of finite type invariants.
  The final chapter is concerned with properties of link groups. In particular
we study the relationship between known solutions from small cancellation
theory and normal surface theory for the word and conjugacy problems of the
groups of (prime) alternating links. We show that two of the algorithms in the
literature for solving the word problem, each using one of the two approaches,
are the same. Then, by considering small cancellation methods, we give a normal
surface solution to the conjugacy problem of these link groups and characterize
the conjugacy classes. Finally as an application of the small cancellation
properties of link groups we give a new proof that alternating links are
non-trivial.
