A knot type is exchange reducible if an arbitrary closed –braid representative of can be changed to a closed braid of minimum braid index by a finite sequence of braid isotopies, exchange moves and –destabilizations. In a preprint of Birman and Wrinkle, a transversal knot in the standard contact structure for 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.
William W Menasco. "On iterated torus knots and transversal knots." Geom. Topol. 5 (2) 651 - 682, 2001. https://doi.org/10.2140/gt.2001.5.651