Geometry & Topology

On iterated torus knots and transversal knots

William W Menasco

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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

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds 57N16: Geometric structures on manifolds [See also 57M50] 57R17: Symplectic and contact topology
Secondary: 37F20: Combinatorics and topology

Keywords
contact structures braids torus knots cabling exchange reducibility

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


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References

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