Geometry & Topology, Vol. 5 (2001) Paper no. 21, pages 651--682.

On iterated torus knots and transversal knots

William W Menasco

Abstract. A knot type is exchange reducible if an arbitrary closed n-braid representative can be changed to a closed braid of minimum braid index by a finite sequence of braid isotopies, exchange moves and +/- destabilizations. In the manuscript [J Birman and NC Wrinkle, On transversally simple knots, preprint (1999)] 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 of Birman and Wrinkle [op cit] 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

AMS subject classification. Primary: 57M27, 57N16, 57R17. Secondary: 37F20.

DOI: 10.2140/gt.2001.5.651

E-print: arXiv:math.GT/0002110

Submitted to GT on 27 March 2001. (Revised 17 July 2001.) Paper accepted 15 August 2001. Paper published 15 August 2001.

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William W Menasco
University at Buffalo, Buffalo, New York 14214, USA
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