#### Volume 5, issue 2 (2001)

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On iterated torus knots and transversal knots

### William W Menasco

Geometry & Topology 5 (2001) 651–682
 arXiv: math.GT/0002110
##### Abstract

A knot type is exchange reducible if an arbitrary closed $n$–braid representative $K$ of $\mathsc{K}$ can be changed to a closed braid of minimum braid index ${n}_{min}\left(\mathsc{K}\right)$ 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 ${S}^{3}$ 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.

##### Keywords
contact structures, braids, torus knots, cabling, exchange reducibility
##### Mathematical Subject Classification 2000
Primary: 57M27, 57N16, 57R17
Secondary: 37F20