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.
Notes on file formats
William W Menasco
University at Buffalo, Buffalo, New York 14214, USA
Email: menasco@tait.math.buffalo.edu
URL: http://www.math.buffalo.edu/~menasco
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