Algebraic & Geometric Topology

An annular refinement of the transverse element in Khovanov homology

Diana Hubbard and Adam Saltz

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

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


We construct a braid conjugacy class invariant κ by refining Plamenevskaya’s transverse element ψ in Khovanov homology via the annular grading. While κ is not an invariant of transverse links, it distinguishes some braids whose closures share the same classical invariants but are not transversely isotopic. Using κ we construct an obstruction to negative destabilization (stronger than ψ) and a solution to the word problem in braid groups. Also, κ is a lower bound on the length of the spectral sequence from annular Khovanov homology to Khovanov homology, and we obtain concrete examples in which this spectral sequence does not collapse immediately. In addition, we study these constructions in reduced Khovanov homology and illustrate that the two reduced versions are fundamentally different with respect to the annular filtration.

Article information

Algebr. Geom. Topol., Volume 16, Number 4 (2016), 2305-2324.

Received: 4 August 2015
Revised: 11 November 2015
Accepted: 4 December 2015
First available in Project Euclid: 28 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F36: Braid groups; Artin groups 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds 57R17: Symplectic and contact topology

Khovanov homology transverse knot invariant braids


Hubbard, Diana; Saltz, Adam. An annular refinement of the transverse element in Khovanov homology. Algebr. Geom. Topol. 16 (2016), no. 4, 2305--2324. doi:10.2140/agt.2016.16.2305.

Export citation


  • E Artin, Theorie der Zöpfe, Abh. Math. Sem. Univ. Hamburg 4 (1925) 47–72
  • E Artin, Theory of braids, Ann. of Math. 48 (1947) 101–126
  • M,M Asaeda, J,H Przytycki, A,S Sikora, Categorification of the Kauffman bracket skein module of $I$–bundles over surfaces, Algebr. Geom. Topol. 4 (2004) 1177–1210
  • 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
  • D Bar-Natan, Khovanov's homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443–1499
  • D Bennequin, Entrelacements et équations de Pfaff, from: “IIIe rencontre de géométrie du Schnepfenried”, Astérisque 2, Société Mathématique de France, Paris (1983) 87–161
  • J,S Birman, N,C Wrinkle, On transversally simple knots, J. Differential Geom. 55 (2000) 325–354
  • P Dehornoy, I Dynnikov, D Rolfsen, B Wiest, Why are braids orderable?, Panoramas et Synthèses 14, Société Mathématique de France, Paris (2002)
  • J,B Etnyre, Legendrian and transversal knots, from: “Handbook of knot theory”, (W Menasco, M Thistlethwaite, editors), Elsevier B. V., Amsterdam (2005) 105–185
  • F,A Garside, The braid group and other groups, Quart. J. Math. Oxford Ser. 20 (1969) 235–254
  • V Gebhardt, J González-Meneses, Solving the conjugacy problem in Garside groups by cyclic sliding, J. Symbolic Comput. 45 (2010) 629–656
  • P Ghiggini, K Honda, J Van Horn-Morris, The vanishing of the contact invariant in the presence of torsion, preprint (2008)
  • J,E Grigsby, Y Ni, Sutured Khovanov homology distinguishes braids from other tangles, Math. Res. Lett. 21 (2014) 1263–1275
  • J,E Grigsby, S,M Wehrli, On gradings in Khovanov homology and sutured Floer homology, from: “Topology and geometry in dimension three”, (W Li, L Bartolini, J Johnson, F Luo, R Myers, J,H Rubinstein, editors), Contemp. Math. 560, Amer. Math. Soc., Providence, RI (2011) 111–128
  • D Hubbard, On sutured Khovanov homology and axis-preserving mutations, preprint (2015)
  • H Hunt, H Keese, A Licata, S Morrison, Computing annular Khovanov homology, preprint (2015)
  • M Hutchings, Introduction to spectral sequences (2011) Available at \setbox0\makeatletter\@url {\unhbox0
  • T Khandhawit, L Ng, A family of transversely nonsimple knots, Algebr. Geom. Topol. 10 (2010) 293–314
  • M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359–426
  • M Khovanov, Patterns in knot cohomology, I, Experiment. Math. 12 (2003) 365–374
  • J Latschev, C Wendl, Algebraic torsion in contact manifolds, Geom. Funct. Anal. 21 (2011) 1144–1195
  • Y-J Lee, Heegaard splittings and Seiberg–Witten monopoles, from: “Geometry and topology of manifolds”, (H,U Boden, I Hambleton, A,J Nicas, B,D Park, editors), Fields Inst. Commun. 47, Amer. Math. Soc., Providence, RI (2005) 173–202
  • R Lipshitz, L Ng, S Sarkar, On transverse invariants from Khovanov homology, preprint (2013)
  • J McCleary, A user's guide to spectral sequences, 2nd edition, Cambridge Studies in Advanced Mathematics 58, Cambridge Univ. Press (2001)
  • S,Y Orevkov, V,V Shevchishin, Markov theorem for transversal links, J. Knot Theory Ramifications 12 (2003) 905–913
  • 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
  • O Plamenevskaya, Transverse knots and Khovanov homology, Math. Res. Lett. 13 (2006) 571–586
  • L,P Roberts, On knot Floer homology in double branched covers, Geom. Topol. 17 (2013) 413–467
  • A Shumakovitch, Torsion of the Khovanov homology, preprint (2004)
  • N Wrinkle, The Markov theorem for transverse knots, PhD thesis, Columbia University (2002) Available at \setbox0\makeatletter\@url {\unhbox0
  • M-L Yau, Vanishing of the contact homology of overtwisted contact 3-manifolds, Bull. Inst. Math. Acad. Sin. 1 (2006) 211–229