On iterated torus knots and transversal knots

Details On iterated torus knots and transversal knots On iterated torus knots and transversal knots

2000-02-14

arxiv.org/abs/math/0002110v2

Geom. Topol. 5(2001) 651-682

Mathematics

Geometric Topology

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper21.abs.html

Scientific paper

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.

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