## Undergraduate Research Projects

This project investigates the applications of knot theory to
chemistry and in particular, whether a moelcule is achiral or chiral,
that is whether or not it can be deformed into its mirror image.
Roughly speaking, we can think of a molecule as being represented by
a graph, and the same applies to knots, so it is possible to apply
knot theory to the problem of chirality of molecules. Also to study
chirality of molecules, the generalization of knots to links is
required at some points.
Note: some figures are missing in the pdf-file.

This paper uses the action of groups acting on
hyperbolic 2 and 3-space to prove that certain groups are unique
product groups. Specifically it is proved that torsion free Fuchsian
groups and torsion free subgroups of Picard's group are unique
product groups. Though these results can be obtained from the known
structure of these groups, it would seem reasonable that the
techniques used in this paper will give new examples of unique
product groups.