On Transversally Simple Knots

Abstract

This paper studies knots that are transversal to the standard contact structure in IR 3 , bringing techniques from topological knot theory to bear on their transversal classification. We say that a transversal knot type T K is transversally simple if it is determined by its topological knot type K and its Bennequin number. The main theorem asserts that any T K whose associated K satisfies a condition that we call exchange reducibility is transversally simple. As applications, we give a new proof of a theorem of Eliashberg [El91], which asserts that the unknot is transversally simple, and (with the help of a new theorem of Menasco [Me99]) extend a result of Etnyre [Et99] to prove that all iterated torus knots are transversally simple. 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 T K is transversally simple. We also give examples of pairs of transversal knots that we conjecture are not transversally simple.