On iterated torus knots and transversal knots

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 $n_{min} (K)$ by a finite sequence of braid isotopies, exchange moves and $\pm$ –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

References

Forward citations

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

Correction: 24 September 2007

Authors

William W Menasco
University at Buffalo Buffalo New York 14214 USA

Volume 5, issue 2 (2001)

William W Menasco