On iterated torus knots and transversal knots

Abstract

A knot type is exchange reducible if an arbitrary closed $n$–braid representative $K$ of $\mathcal{K}$ can be changed to a closed braid of minimum braid index $n_{min}\left(\mathcal{K}\right)$ by a finite sequence of braid isotopies, exchange moves and $\pm$–destabilizations. In a preprint of Birman and Wrinkle, a transversal knot in the standard contact structure for $S^3$ is defined to be transversally simple if it is characterized up to transversal isotopy by its topological knot type and its self-linking number. Theorem 2 in the preprint establishes that exchange reducibility implies transversally simplicity. The main result in this note, establishes that iterated torus knots are exchange reducible. It then follows as a corollary that iterated torus knots are transversally simple.