Volume 5, issue 2 (2001)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
On iterated torus knots and transversal knots

William W Menasco

Geometry & Topology 5 (2001) 651–682

Erratum: Geometry & Topology 11 (2007) 1731–1732

arXiv: math.GT/0002110

Abstract

A knot type is exchange reducible if an arbitrary closed n–braid representative K of K can be changed to a closed braid of minimum braid index nmin(K) 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 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 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
References
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Publication
Received: 27 March 2001
Revised: 17 July 2001
Accepted: 15 August 2001
Published: 15 August 2001
Proposed: Yasha Eliashberg
Seconded: Robion Kirby, Joan Birman
Authors
William W Menasco
University at Buffalo
Buffalo
New York 14214
USA