Transverse link invariants from the deformations of Khovanov 𝔰𝔩3–homology

Collari, Carlo¹

¹Alfréd Rényi Institute of Mathematics, Budapest, Hungary, New York University Abu Dhabi, Saadiyat Island, Abu Dhabi, United Arab Emirates

Algebraic and Geometric Topology, Tome 20 (2020) no. 4, pp. 1729-1768

Résumé

We make use of the Mackaay–Vaz approach to the universal 𝔰𝔩3 –homology to define a family of cycles (called β3 –invariants) which are transverse braid invariants. This family includes Wu’s ψ3 –invariant. Furthermore, we analyse the vanishing of the homology classes of the β3 –invariants and relate it to the vanishing of Plamenevskaya’s and Wu’s invariants. Finally, we use the β3 –invariants to produce some Bennequin-type inequalities.

Détail

DOI : 10.2140/agt.2020.20.1729

Classification : 57M25, 57R17, 57M27
Keywords: transverse invariants in $S^3$, Khovanov $\mathrm{sl}(3)$ homology, Plamenevskaya invariant

Bibliographie

[1] D Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443 | DOI

[2] D Bennequin, Entrelacements et équations de Pfaff, from: "Third Schnepfenried geometry conference, I", Astérisque 107, Soc. Math. France, Paris (1983) 87

[3] C Collari, Transverse invariants from Khovanov-type homologies, J. Knot Theory Ramifications 28 (2019) | DOI

[4] J B Etnyre, Legendrian and transversal knots, from: "Handbook of knot theory" (editors W Menasco, M Thistlethwaite), Elsevier, Amsterdam (2005) 105 | DOI

[5] B Gornik, Note on Khovanov link cohomology, preprint (2004)

[6] M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359 | DOI

[7] M Khovanov, Patterns in knot cohomology, I, Experiment. Math. 12 (2003) 365 | DOI

[8] M Khovanov, L Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008) 1 | DOI

[9] G Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 (1996) 109 | DOI

[10] S Lang, Algebra, 211, Springer (2002) | DOI

[11] L Lewark, Rasmussen’s spectral sequences and the slN–concordance invariants, Adv. Math. 260 (2014) 59 | DOI

[12] L Lewark, A Lobb, New quantum obstructions to sliceness, Proc. Lond. Math. Soc. 112 (2016) 81 | DOI

[13] L Lewark, A Lobb, Upsilon-like concordance invariants from sln knot cohomology, Geom. Topol. 23 (2019) 745 | DOI

[14] R Lipshitz, L Ng, S Sarkar, On transverse invariants from Khovanov homology, Quantum Topol. 6 (2015) 475 | DOI

[15] C Livingston, S Naik, Ozsváth–Szabó and Rasmussen invariants of doubled knots, Algebr. Geom. Topol. 6 (2006) 651 | DOI

[16] A Lobb, A note on Gornik’s perturbation of Khovanov–Rozansky homology, Algebr. Geom. Topol. 12 (2012) 293 | DOI

[17] M Mackaay, P Vaz, The universal sl3–link homology, Algebr. Geom. Topol. 7 (2007) 1135 | DOI

[18] M Mackaay, P Vaz, The foam and the matrix factorization sl3 link homologies are equivalent, Algebr. Geom. Topol. 8 (2008) 309 | DOI

[19] S Y Orevkov, V V Shevchishin, Markov theorem for transversal links, J. Knot Theory Ramifications 12 (2003) 905 | DOI

[20] O Plamenevskaya, Transverse knots and Khovanov homology, Math. Res. Lett. 13 (2006) 571 | DOI

[21] M Polyak, Minimal generating sets of Reidemeister moves, Quantum Topol. 1 (2010) 399 | DOI

[22] D E V Rose, P Wedrich, Deformations of colored slN link homologies via foams, Geom. Topol. 20 (2016) 3431 | DOI

[23] N C Wrinkle, The Markov theorem for transverse knots, PhD thesis, Columbia University (2002)

[24] H Wu, Braids, transversal links and the Khovanov–Rozansky theory, Trans. Amer. Math. Soc. 360 (2008) 3365 | DOI

[25] H Wu, On the quantum filtration of the Khovanov–Rozansky cohomology, Adv. Math. 221 (2009) 54 | DOI