Transverse invariants from sl_3 homologies

Carlo Collari (Università di Pisa)

11 dec 2018 -- 12:00

Aula Tricerri, DiMaI, Firenze

Abstract.

In this talk I will describe some invariants for transverse links in S³ (endowed with the symmetric contact structure) arising from the deformations of Khovanov sl₃ homology.

I will start with a brief introduction to the theory of transverse links in S^3. Afterward, I will recall some known results concerning transverse invariants in link homologies. In particular, I will focus on the invariants coming from Khovanov-Rozansky homologies, and those coming from the deformations of Khovanov homology.

After the introductory part, I will briefly describe the construction and the properties of the universal sl₃ link homology, due to Mackaay and Vaz, and define the transverse invariants. These invariants, which are called the \beta₃-invariants, are cycles in the universal sl₃ complex.

Finally, I will state a result concerning the vanishing of the homology classes of these invariants, and compare it with similar results known for other invariants. The idea of the proof will be also given. Time permitting, I will also give a number of Bennequin-type inequalities which can be proved using the \beta₃-invariants.