Volume 20, issue 4 (2020)

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Transverse link invariants from the deformations of Khovanov $\mathfrak{sl}_{3}$–homology

Carlo Collari

Algebraic & Geometric Topology 20 (2020) 1729–1768
Abstract

We make use of the Mackaay–Vaz approach to the universal ${\mathfrak{𝔰}\mathfrak{𝔩}}_{3}$–homology to define a family of cycles (called ${\beta }_{3}$–invariants) which are transverse braid invariants. This family includes Wu’s ${\psi }_{3}$–invariant. Furthermore, we analyse the vanishing of the homology classes of the ${\beta }_{3}$–invariants and relate it to the vanishing of Plamenevskaya’s and Wu’s invariants. Finally, we use the ${\beta }_{3}$–invariants to produce some Bennequin-type inequalities.

Keywords
transverse invariants in $S^3$, Khovanov $\mathrm{sl}(3)$ homology, Plamenevskaya invariant
Mathematical Subject Classification 2010
Primary: 57M25, 57R17
Secondary: 57M27