## Geometry & Topology

### On iterated torus knots and transversal knots

William W Menasco

#### 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.

#### Article information

Source
Geom. Topol., Volume 5, Number 2 (2001), 651-682.

Dates
Received: 27 March 2001
Revised: 17 July 2001
Accepted: 15 August 2001
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883040

Digital Object Identifier
doi:10.2140/gt.2001.5.651

Mathematical Reviews number (MathSciNet)
MR1857523

Zentralblatt MATH identifier
1002.57025

#### Citation

Menasco, William W. On iterated torus knots and transversal knots. Geom. Topol. 5 (2001), no. 2, 651--682. doi:10.2140/gt.2001.5.651. https://projecteuclid.org/euclid.gt/1513883040

#### References

• J Birman, E Finkelstein, Studying surfaces via closed braids, Journal of Knot Theory and its Ramifications, 7 (1998) 267–334
• J Birman, W Menasco, Studying Links via Closed Braids V: Closed braid representations of the unlink, Transactions of the AMS, 329 (1992) 585–606
• J Birman, W Menasco, Studying Links via Closed Braids IV: Split links and composite links, Inventiones Mathematicae, 102 (1990) 115–139
• J Birman, W Menasco, Special positions for essential tori in link complements, Topology, 33 (1994) 525–556
• J Birman, W Menasco, Studying Links via Closed Braids IV: Closed links which are $3$–braids, Pacific Journal of Mathematics, 161 (1993) 25–113
• J Birman, N C Wrinkle, On transversally simple knots, preprint (1999)
• Y Eliashberg, Legendrian and transversal knots in tight contact $3$–manifolds, Topological Methods in Modern Mathematics, (1991) 171–193
• J Etnyre, Transversal torus knots, Geometry and Topology, 3 (1999) 253–268
• W Jaco, Lectures on Three–Manifold Topology, AMS Regional conference series, No. 43
• J Los, Knots, braid index and dynamical type, Topology, 33 (1994) 257–270
• D Rolfsen, Knots and Links, Mathematics Lecture Series 7, Publish or Perish, Inc. (1976)
• H Schubert, Knoten und Vollringe, Acta Math. 90 (1953) 131–226