## Algebraic & Geometric Topology

### A family of transverse link homologies

Hao Wu

#### Abstract

We define a homology $ℋN$ for closed braids by applying Khovanov and Rozansky’s matrix factorization construction with potential $axN+1$. Up to a grading shift, $ℋ0$ is the HOMFLYPT homology defined by Khovanov and Rozansky. We demonstrate that for $N ≥ 1$, $ℋN$ is a $ℤ2 ⊕ ℤ⊕3$–graded $ℚ[a]$–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 $ℋN$ and the relation between $ℋN$ and the $sl(N)$ Khovanov–Rozansky homology.

#### Article information

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

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

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

#### 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

#### 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