The Two-Color Ext Soergel Calculus

TL;DR

This paper computes Ext groups for Soergel Bimodules in dihedral groups, providing explicit bases and diagrammatic presentations, and applies these results to link homology and categorification of Gomi's trace.

Contribution

It introduces a diagrammatic basis for Hochschild cohomology of Soergel Bimodules and presents a monoidal category of Ext-enhanced bimodules, advancing the understanding of their structure.

Findings

Explicit diagrammatic basis for Hochschild cohomology.

Computed HOMFLY homology for specific links.

Categorifies Gomi's trace via Hochschild cohomology.

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.