For any braid, the outermost three T-degree layers of reduced triply-graded HOMFLY homology are determined explicitly, and are almost entirely zero with only a few one-dimensional k-modules.
The Two-Color Ext Soergel Calculus
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abstract
We compute Ext groups between Soergel Bimodules associated to the infinite/finite dihedral group for a realization in characteristic 0 and show that they are free right $R-$modules. In particular, we obtain an explicit diagrammatic basis for the Hochschild cohomology of indecomposable Soergel Bimodules. We then give a diagrammatic presentation for the corresponding monoidal category of Ext-enhanced Soergel Bimodules. As applications, we explicitly compute HOMFLY homology/triply graded link homology $\overline{\mathrm{HHH}}$ for the connect sum of two Hopf links and the negative torus link $T(3,-3)$ as right $R-$modules. Furthermore, we show that the Hochschild cohomology of Soergel Bimodules in finite dihedral type categorifies Gomi's trace, providing a $t-$analog of Soergel's Hom Formula in the dihedral setting.
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The Last Three T-degrees in Triply-Graded Link Homology
For any braid, the outermost three T-degree layers of reduced triply-graded HOMFLY homology are determined explicitly, and are almost entirely zero with only a few one-dimensional k-modules.