Algebraic & Geometric Topology

Quasi-right-veering braids and nonloose links

Tetsuya Ito and Keiko Kawamuro

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Abstract

We introduce a notion of quasi-right-veering for closed braids, which plays an analogous role to right-veering for open books. We show that a transverse link K in a contact 3–manifold (M,ξ) is nonloose if and only if every braid representative of K with respect to every open book decomposition that supports (M,ξ) is quasi-right-veering. We also show that several definitions of right-veering closed braids are equivalent.

Article information

Source
Algebr. Geom. Topol., Volume 19, Number 6 (2019), 2989-3032.

Dates
Received: 15 May 2018
Revised: 10 December 2018
Accepted: 30 January 2019
First available in Project Euclid: 29 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.agt/1572314548

Digital Object Identifier
doi:10.2140/agt.2019.19.2989

Mathematical Reviews number (MathSciNet)
MR4023334

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 57M27: Invariants of knots and 3-manifolds

Keywords
quasi-right-veering loose transverse knots

Citation

Ito, Tetsuya; Kawamuro, Keiko. Quasi-right-veering braids and nonloose links. Algebr. Geom. Topol. 19 (2019), no. 6, 2989--3032. doi:10.2140/agt.2019.19.2989. https://projecteuclid.org/euclid.agt/1572314548


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References

  • K L Baker, S Onaran, Nonlooseness of nonloose knots, Algebr. Geom. Topol. 15 (2015) 1031–1066
  • J A Baldwin, J E Grigsby, Categorified invariants and the braid group, Proc. Amer. Math. Soc. 143 (2015) 2801–2814
  • J A Baldwin, D S Vela-Vick, V Vértesi, On the equivalence of Legendrian and transverse invariants in knot Floer homology, Geom. Topol. 17 (2013) 925–974
  • D Bennequin, Entrelacements et équations de Pfaff, from “Third Schnepfenried geometry conference, I”, Astérisque 107, Soc. Mat. de 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, Stabilization in the braid groups, II: Transversal simplicity of knots, Geom. Topol. 10 (2006) 1425–1452
  • V Colin, K Honda, Reeb vector fields and open book decompositions, J. Eur. Math. Soc. 15 (2013) 443–507
  • Y Eliashberg, Classification of overtwisted contact structures on $3$–manifolds, Invent. Math. 98 (1989) 623–637
  • Y Eliashberg, M Fraser, Topologically trivial Legendrian knots, J. Symplectic Geom. 7 (2009) 77–127
  • J Epstein, D Fuchs, M Meyer, Chekanov–Eliashberg invariants and transverse approximations of Legendrian knots, Pacific J. Math. 201 (2001) 89–106
  • J B Etnyre, Lectures on open book decompositions and contact structures, from “Floer homology, gauge theory, and low-dimensional topology” (D A Ellwood, P S Ozsváth, A I Stipsicz, Z Szabó, editors), Clay Math. Proc. 5, Amer. Math. Soc., Providence, RI (2006) 103–141
  • J B Etnyre, On knots in overtwisted contact structures, Quantum Topol. 4 (2013) 229–264
  • J B Etnyre, K Honda, Knots and contact geometry, I: Torus knots and the figure eight knot, J. Symplectic Geom. 1 (2001) 63–120
  • J B Etnyre, D S Vela-Vick, Torsion and open book decompositions, Int. Math. Res. Not. 2010 (2010) 4385–4398
  • B Farb, D Margalit, A primer on mapping class groups, Princeton Mathematical Series 49, Princeton Univ. Press (2012)
  • E Giroux, Convexité en topologie de contact, Comment. Math. Helv. 66 (1991) 637–677
  • E Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, from “Proceedings of the International Congress of Mathematicians” (T Li, editor), volume 2, Higher Ed. Press, Beijing (2002) 405–414
  • K Honda, W H Kazez, G Matić, Right-veering diffeomorphisms of compact surfaces with boundary, Invent. Math. 169 (2007) 427–449
  • K Honda, W H Kazez, G Matić, Right-veering diffeomorphisms of compact surfaces with boundary, II, Geom. Topol. 12 (2008) 2057–2094
  • T Ito, K Kawamuro, Open book foliation, Geom. Topol. 18 (2014) 1581–1634
  • T Ito, K Kawamuro, Operations on open book foliations, Algebr. Geom. Topol. 14 (2014) 2983–3020
  • T Ito, K Kawamuro, Visualizing overtwisted discs in open books, Publ. Res. Inst. Math. Sci. 50 (2014) 169–180
  • T Ito, K Kawamuro, Overtwisted discs in planar open books, Internat. J. Math. 26 (2015) art. id. 1550027
  • T Ito, K Kawamuro, Coverings of open books, from “Advances in the mathematical sciences” (G Letzter, K Lauter, E Chambers, N Flournoy, J E Grigsby, C Martin, K Ryan, K Trivisa, editors), Assoc. Women Math. Ser. 6, Springer (2016) 139–154
  • T Ito, K Kawamuro, Essential open book foliations and fractional Dehn twist coefficient, Geom. Dedicata 187 (2017) 17–67
  • T Ito, K Kawamuro, The defect of Bennequin–Eliashberg inequality and Bennequin surfaces, Indiana Univ. Math. J. 68 (2019) 799–833
  • D J LaFountain, W W Menasco, Braid foliations in low-dimensional topology, Graduate Studies in Mathematics 185, Amer. Math. Soc., Providence, RI (2017)
  • Y Mitsumatsu, A Mori, On Bennequin's isotopy lemma Appendix to Y Mitsumatsu, Convergence of contact structures to foliations, from “Foliations 2005” (P Walczak, R Langevin, S Hurder, T Tsuboi, editors), World Sci., Hackensack, NJ (2006) 353–371
  • E Pavelescu, Braids and open book decompositions, PhD thesis, University of Pennsylvania (2008) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/304492097 {\unhbox0
  • E Pavelescu, Braiding knots in contact $3$–manifolds, Pacific J. Math. 253 (2011) 475–487
  • O Plamenevskaya, Braid monodromy, orderings and transverse invariants, Algebr. Geom. Topol. 18 (2018) 3691–3718
  • W P Thurston, H E Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975) 345–347