On iterated torus knots and transversal knots

A knot type is exchange reducible if an arbitrary closed n-braid representative K of Κ can be changed to a closed braid of minimum braid index n_min(Κ) by a finite sequence of braid isotopies, exchange moves and ±-destabilizations. (See Figure 1). In the manuscript [6] a transversal knot in the standard contact structure for S³ 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 of [6] establishes that exchange reducibility implies transversally simplicity. Theorem 1.1, 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.