Algebraic & Geometric Topology

A family of transverse link homologies

Hao Wu

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

We define a homology N for closed braids by applying Khovanov and Rozansky’s matrix factorization construction with potential axN+1. Up to a grading shift, 0 is the HOMFLYPT homology defined by Khovanov and Rozansky. We demonstrate that for N 1, N is a 2 3–graded [a]–module that is invariant under transverse Markov moves, but not under negative stabilization/destabilization. Thus, for N 1, this homology is an invariant for transverse links in the standard contact S3, but not for smooth links. We also discuss the decategorification of N and the relation between N and the sl(N) Khovanov–Rozansky homology.

Article information

Source
Algebr. Geom. Topol., Volume 16, Number 1 (2016), 41-127.

Dates
Received: 8 April 2014
Revised: 12 February 2015
Accepted: 15 April 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841104

Digital Object Identifier
doi:10.2140/agt.2016.16.41

Mathematical Reviews number (MathSciNet)
MR3470697

Zentralblatt MATH identifier
1345.57018

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57R17: Symplectic and contact topology

Keywords
transverse link Khovanov–Rozansky homology HOMFLYPT polynomial

Citation

Wu, Hao. A family of transverse link homologies. Algebr. Geom. Topol. 16 (2016), no. 1, 41--127. doi:10.2140/agt.2016.16.41. https://projecteuclid.org/euclid.agt/1510841104


Export citation

References

  • D Bar-Natan, Fast Khovanov homology computations, J. Knot Theory Ramifications 16 (2007) 243–255
  • D Bennequin, Entrelacements et équations de Pfaff, from: “Third Schnepfenried geometry conference, I”, Astérisque 107, Soc. Math. France, Paris (1983) 87–161
  • D,J Benson, Representations and cohomology, I: Basic representation theory of finite groups and associative algebras, Cambridge Studies in Advanced Mathematics 30, Cambridge Univ. Press (1991)
  • J,S Birman, W,W Menasco, Stabilization in the braid groups, II: transversal simplicity of knots, Geom. Topol. 10 (2006) 1425–1452
  • J,B Etnyre, Introductory lectures on contact geometry, from: “Topology and geometry of manifolds”, (G Matić, C McCrory, editors), Proc. Sympos. Pure Math. 71, Amer. Math. Soc. (2003) 81–107
  • J,B Etnyre, K Honda, Cabling and transverse simplicity, Ann. of Math. 162 (2005) 1305–1333
  • J Franks, R,F Williams, Braids and the Jones polynomial, Trans. Amer. Math. Soc. 303 (1987) 97–108
  • B Gornik, Note on Khovanov link cohomology, preprint
  • M Khovanov, L Rozansky, Virtual crossings, convolutions and a categorification of the ${\rm SO}(2N)$ Kauffman polynomial, J. Gökova Geom. Topol. 1 (2007) 116–214
  • M Khovanov, L Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008) 1–91
  • M Khovanov, L Rozansky, Matrix factorizations and link homology, II, Geom. Topol. 12 (2008) 1387–1425
  • D Krasner, Equivariant ${\rm sl}(n)$–link homology, Algebr. Geom. Topol. 10 (2010) 1–32
  • A Lascoux, Interpolation, notes (1996) Available at \setbox0\makeatletter\@url http://tinyurl.com/lascoux-interp {\unhbox0
  • L Ng, A topological introduction to knot contact homology, from: “Contact and symplectic topology”, (F Bourgeois, V Colin, A Stipsicz, editors), Bolyai Soc. Math. Stud. 26, János Bolyai Math. Soc., Budapest (2014) 485–530
  • S,Y Orevkov, V,V Shevchishin, Markov theorem for transversal links, J. Knot Theory Ramifications 12 (2003) 905–913
  • D,S Passman, A course in ring theory, Wadsworth & Brooks, Pacific Grove, CA (1991)
  • J Rasmussen, Some differentials on Khovanov–Rozansky homology, Geom. Topol. 19 (2015) 3031–3104
  • N,C Wrinkle, The Markov theorem for transverse knots, PhD thesis, Columbia University (2002) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/304800949 {\unhbox0
  • H Wu, On the transverse Khovanov–Rozansky homologies: graded module structure and stabilization
  • H Wu, On the quantum filtration of the Khovanov–Rozansky cohomology, Adv. Math. 221 (2009) 54–139
  • H Wu, A colored $\mathfrak{sl}(N)$–homology for links in $S\sp 3$, Dissertationes Math. $($Rozprawy Mat.$)$ 499 (2014) 217
  • H Wu, Equivariant Khovanov–Rozansky homology and Lee–Gornik spectral sequence, Quantum Topol. 6 (2015) 515–607