The Last Three T-degrees in Triply-Graded Link Homology

We investigate the structure of reduced triply graded link homology $\overline{\mathrm{H}\mathrm{H}\mathrm{H}}$ in the top/bottom three $T-$degrees for links arising as closures of positive/negative braids. Using a diagrammatic approach to the Hochschild cohomology of Soergel bimodules, we provide explicit computations of $\overline{\mathrm{H}\mathrm{H}\mathrm{H}}$ as $R-$modules in these degrees. Our results reveal that the homology here is often zero, especially in the negative braid case, and display striking uniformity.