Geometry & Topology

On iterated torus knots and transversal knots

William W Menasco

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

contact structures braids torus knots cabling exchange reducibility


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.

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  • 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