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 $TK$ is {\it transversally simple} if it is determined by its topological knot type $K$ and its Bennequin number. The main theorem asserts that any $TK$ whose associated $K$ 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 [El]. As a second application we use a new theorem of Menasco [Me] to extend a result of Etnyre [Et] to prove that all 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 $K$ in order to prove that any associated $TK$ is transversally simple. We also give examples of pairs of transversal knots that we conjecture are {\em not} transversally simple.