  Final revision. To appear in the Journal of Differential Geometry. This paper
studies knots that are transversal to the standard contact structure in
$\reals^3$, bringing techniques from topological knot theory to bear on their
transversal classification. We say that a transversal knot type $\cTK$ is {\it
transversally simple} if it is determined by its topological knot type $\cK$
and its Bennequin number. The main theorem asserts that any $\cTK$ whose
associated $\cK$ satisfies a condition that we call {\em exchange reducibility}
is transversally simple.
  As a first application, we prove that the unlink is transversally simple,
extending the main theorem in \cite{El}. As a second application we use a new
theorem of Menasco (Theorem 1 of \cite{Me}) to extend a result of Etnyre
\cite{Et} to prove that iterated torus knots are transversally simple. We also
give a formula for their maximum Bennequin number. We show that the concept of
exchange reducibility is the simplest of the constraints that one can place on
$\cK$ in order to prove that any associated $\cTK$ is transversally simple. We
also give examples of pairs of transversal knots that we conjecture are {\em
not} transversally simple.
