## Geometry & Topology

### Stabilization in the braid groups II: Transversal simplicity of knots

#### Abstract

The main result of this paper is a negative answer to the question: are all transversal knot types transversally simple? An explicit infinite family of examples is given of closed 3–braids that define transversal knot types that are not transversally simple. The method of proof is topological and indirect.

#### Article information

Source
Geom. Topol., Volume 10, Number 3 (2006), 1425-1452.

Dates
Accepted: 28 June 2006
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799772

Digital Object Identifier
doi:10.2140/gt.2006.10.1425

Mathematical Reviews number (MathSciNet)
MR2255503

Zentralblatt MATH identifier
1130.57005

#### Citation

Birman, Joan S; Menasco, William W. Stabilization in the braid groups II: Transversal simplicity of knots. Geom. Topol. 10 (2006), no. 3, 1425--1452. doi:10.2140/gt.2006.10.1425. https://projecteuclid.org/euclid.gt/1513799772

#### References

• V I Arnol'd, Topological invariants of plane curves and caustics, University Lecture Series 5, American Mathematical Society, Providence, RI (1994)
• D Bennequin, Entrelacements et équations de Pfaff, from: “Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982)”, Astérisque 107, Soc. Math. France, Paris (1983) 87–161
• J S Birman, E Finkelstein, Studying surfaces via closed braids, J. Knot Theory Ramifications 7 (1998) 267–334
• J S Birman, W W Menasco, Studying links via closed braids. III. Classifying links which are closed $3$-braids, Pacific J. Math. 161 (1993) 25–113
• J S Birman, W W Menasco, On Markov's theorem, J. Knot Theory Ramifications 11 (2002) 295–310
• J S Birman, W W Menasco, Stabilization in the braid groups. I. MTWS, Geom. Topol. 10 (2006) 413–540
• J S Birman, N C Wrinkle, On transversally simple knots, J. Differential Geom. 55 (2000) 325–354
• Y Eliashberg, Legendrian and transversal knots in tight contact $3$-manifolds, from: “Topological methods in modern mathematics (Stony Brook, NY, 1991)”, Publish or Perish, Houston, TX (1993) 171–193
• J B Etnyre, K Honda, Cabling and transverse simplicity, Ann. of Math. $(2)$ 162 (2005) 1305–1333
• D Fuchs, S Tabachnikov, Invariants of Legendrian and transverse knots in the standard contact space, Topology 36 (1997) 1025–1053
• E Giroux, Géométrie de Contact:de la Dimension Trois vers les Dimensions Supérieures
• K H Ko, S J Lee, Flypes of closed $3$-braids in the standard contact space, J. Korean Math. Soc. 36 (1999) 51–71
• H R Morton, Infinitely many fibred knots having the same Alexander polynomial, Topology 17 (1978) 101–104
• K Murasugi, On closed $3$-braids, American Mathematical Society, Providence, R.I. (1974)
• S Y Orevkov, V V Shevchishin, Markov theorem for transversal links, J. Knot Theory Ramifications 12 (2003) 905–913
• N Wrinkle, PhD thesis, Columbia University (2002)