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.