On transversally simple knots

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.