2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/102151We establish some inequalities about the Khovanov-Rozansky cohomologies of braids. These give new upper bounds of the self-linking numbers of transversal links in standard contact $S^3$ which is sharper than the well known bound given by the HOMFLY polynomial. We also introduce a sequence of transversal link invariants, and discuss some of their properties.26 pages, 15 figures. Third version: modified proofs of Lemma 3.1 and Proposition 4.2, also corrected some typos and inconsistency in conventionsGeometric TopologySymplectic Geometry57M25, 57R17text