Algebraic & Geometric Topology

Braid monodromy, orderings and transverse invariants

Olga Plamenevskaya

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/agt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A closed braid β naturally gives rise to a transverse link K in the standard contact 3–space. We study the effect of the dynamical properties of the monodromy of β, such as right-veering, on the contact-topological properties of K and the values of transverse invariants in Heegaard Floer and Khovanov homologies. Using grid diagrams and the structure of Dehornoy’s braid ordering, we show that θ̂(K)HFK̂(m(K)) is nonzero whenever β has fractional Dehn twist coefficient C>1. (For a 3–braid, we get a sharp result: θ̂0 if and only if the braid is right-veering.)

Article information

Source
Algebr. Geom. Topol., Volume 18, Number 6 (2018), 3691-3718.

Dates
Received: 21 March 2018
Revised: 8 June 2018
Accepted: 18 June 2018
First available in Project Euclid: 27 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.agt/1540605654

Digital Object Identifier
doi:10.2140/agt.2018.18.3691

Mathematical Reviews number (MathSciNet)
MR3868232

Zentralblatt MATH identifier
06990075

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds 57R17: Symplectic and contact topology 57R58: Floer homology

Keywords
Heegaard Floer transverse invariants right-veering fractional Dehn twist coefficient grid diagrams

Citation

Plamenevskaya, Olga. Braid monodromy, orderings and transverse invariants. Algebr. Geom. Topol. 18 (2018), no. 6, 3691--3718. doi:10.2140/agt.2018.18.3691. https://projecteuclid.org/euclid.agt/1540605654


Export citation

References

  • J A Baldwin, Heegaard Floer homology and genus one, one-boundary component open books, J. Topol. 1 (2008) 963–992
  • J A Baldwin, Comultiplication in link Floer homology and transversely nonsimple links, Algebr. Geom. Topol. 10 (2010) 1417–1436
  • J A Baldwin, J E Grigsby, Categorified invariants and the braid group, Proc. Amer. Math. Soc. 143 (2015) 2801–2814
  • J A Baldwin, O Plamenevskaya, Khovanov homology, open books, and tight contact structures, Adv. Math. 224 (2010) 2544–2582
  • 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
  • A Beliakova, S Wehrli, Categorification of the colored Jones polynomial and Rasmussen invariant of links, Canad. J. Math. 60 (2008) 1240–1266
  • J S Birman, W W Menasco, A note on closed $3$–braids, Commun. Contemp. Math. 10 (2008) 1033–1047
  • M Casey, Branched covers of contact $3$–manifolds, PhD thesis, Georgia Institute of Technology (2013) Available at \setbox0\makeatletter\@url http://hdl.handle.net/1853/50313 {\unhbox0
  • A Cavallo, The concordance invariant tau in link grid homology, Algebr. Geom. Topol. 18 (2018) 1917–1951
  • P Dehornoy, Braid groups and left distributive operations, Trans. Amer. Math. Soc. 345 (1994) 115–150
  • P Dehornoy, I Dynnikov, D Rolfsen, B Wiest, Ordering braids, Mathematical Surveys and Monographs 148, Amer. Math. Soc., Providence, RI (2008)
  • Y Eliashberg, Classification of overtwisted contact structures on $3$–manifolds, Invent. Math. 98 (1989) 623–637
  • J B Etnyre, K Honda, Cabling and transverse simplicity, Ann. of Math. 162 (2005) 1305–1333
  • R Fenn, M T Greene, D Rolfsen, C Rourke, B Wiest, Ordering the braid groups, Pacific J. Math. 191 (1999) 49–74
  • D Gabai, U Oertel, Essential laminations in $3$–manifolds, Ann. of Math. 130 (1989) 41–73
  • P Ghiggini, Ozsváth–Szabó invariants and fillability of contact structures, Math. Z. 253 (2006) 159–175
  • S Harvey, K Kawamuro, O Plamenevskaya, On transverse knots and branched covers, Int. Math. Res. Not. 2009 (2009) 512–546
  • 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
  • K Honda, W H Kazez, G Matić, On the contact class in Heegaard Floer homology, J. Differential Geom. 83 (2009) 289–311
  • T Ito, Braid ordering and knot genus, J. Knot Theory Ramifications 20 (2011) 1311–1323
  • T Ito, Braid ordering and the geometry of closed braid, Geom. Topol. 15 (2011) 473–498
  • T Ito, K Kawamuro, Quasi right-veering braids and non-loose links, preprint (2016)
  • T Ito, K Kawamuro, Essential open book foliations and fractional Dehn twist coefficient, Geom. Dedicata 187 (2017) 17–67
  • T Khandhawit, L Ng, A family of transversely nonsimple knots, Algebr. Geom. Topol. 10 (2010) 293–314
  • R Lipshitz, L Ng, S Sarkar, On transverse invariants from Khovanov homology, Quantum Topol. 6 (2015) 475–513
  • P Lisca, P Ozsváth, A I Stipsicz, Z Szabó, Heegaard Floer invariants of Legendrian knots in contact three-manifolds, J. Eur. Math. Soc. 11 (2009) 1307–1363
  • A V Malyutin, Twist number of (closed) braids, Algebra i Analiz 16 (2004) 59–91 In Russian; translated in St. Petersburg Math. J. 16 (2005) 791–813
  • A V Malyutin, N Y Netsvetaev, Dehornoy order in the braid group and transformations of closed braids, Algebra i Analiz 15 (2003) 170–187 In Russian; translated in St. Petersburg Math. J. 15 (2004) 437–448
  • C Manolescu, P Ozsváth, S Sarkar, A combinatorial description of knot Floer homology, Ann. of Math. 169 (2009) 633–660
  • K Murasugi, On closed $3$–braids, Mem. Amer. Math. Soc. 151, Amer. Math. Soc., Providence, RI (1974)
  • L Ng, P Ozsváth, D Thurston, Transverse knots distinguished by knot Floer homology, J. Symplectic Geom. 6 (2008) 461–490
  • L Ng, D Thurston, Grid diagrams, braids, and contact geometry, from “Proceedings of Gökova Geometry–Topology Conference 2008” (S Akbulut, T Önder, R J Stern, editors), GGT, Gökova (2009) 120–136
  • S Y Orevkov, Markov moves for quasipositive braids, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000) 557–562
  • S Y Orevkov, V V Shevchishin, Markov theorem for transversal links, J. Knot Theory Ramifications 12 (2003) 905–913
  • P Ozsváth, Z Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004) 311–334
  • P Ozsváth, Z Szabó, Heegaard Floer homology and contact structures, Duke Math. J. 129 (2005) 39–61
  • P Ozsváth, Z Szabó, On the Heegaard Floer homology of branched double-covers, Adv. Math. 194 (2005) 1–33
  • P Ozsváth, Z Szabó, D Thurston, Legendrian knots, transverse knots and combinatorial Floer homology, Geom. Topol. 12 (2008) 941–980
  • O Plamenevskaya, Bounds for the Thurston–Bennequin number from Floer homology, Algebr. Geom. Topol. 4 (2004) 399–406
  • O Plamenevskaya, Transverse knots and Khovanov homology, Math. Res. Lett. 13 (2006) 571–586
  • O Plamenevskaya, Transverse knots, branched double covers and Heegaard Floer contact invariants, J. Symplectic Geom. 4 (2006) 149–170
  • V V Prasolov, A B Sossinsky, Knots, links, braids and $3$–manifolds: an introduction to the new invariants in low-dimensional topology, Translations of Mathematical Monographs 154, Amer. Math. Soc., Providence, RI (1997)
  • J Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 (2010) 419–447
  • L P Roberts, On knot Floer homology in double branched covers, Geom. Topol. 17 (2013) 413–467
  • L Rudolph, Quasipositivity as an obstruction to sliceness, Bull. Amer. Math. Soc. 29 (1993) 51–59
  • A N Shumakovitch, Rasmussen invariant, slice-Bennequin inequality, and sliceness of knots, J. Knot Theory Ramifications 16 (2007) 1403–1412
  • N C Wrinkle, The Markov theorem for transverse knots, PhD thesis, Columbia University (2002) Available at \setbox0\makeatletter\@url https://search.proquest.com/docview/304800949 {\unhbox0