We show that a certain category of bimodules over a finite-dimensional quiver algebra known as a type B zigzag algebra is a quotient category of the category of type B Soergel bimodules. This leads to an alternate proof of Rouquier’s conjecture on the faithfulness of the 2-braid groups for type B.

For a Coxeter system and a representation $V$ of this Coxeter system, Soergel defined a category which is now called the category of Soergel bimodules and proved that this gives a categorification of the Hecke algebra when $V$ is reflection faithful. Elias and Williamson defined another category when $V$ is not reflection faithful and proved that this category is equivalent to the category of Soergel bimodules when $V$ is reflection faithful. Moreover, they proved the categorification theorem for their category with fewer assumptions on $V$. In this paper, we give a bimodule description of the Elias–Williamson category and re-prove the categorification theorem.

We introduce a new method for computing triply graded link homology, which is particularly well adapted to torus links. Our main application is to the $(n,n)$-torus links, for which we give an exact answer for all $n$. In several cases, our computations verify conjectures of Gorsky et al. relating homology of torus links with Hilbert schemes.

We give a diagrammatic presentation for the category of Soergel bimodules for the dihedral group $W$. The (two-colored) Temperley–Lieb category is embedded inside this category as the degree $0$ morphisms between color-alternating objects. The indecomposable Soergel bimodules are the images of Jones–Wenzl projectors. When $W$ is infinite, the parameter $q$ of the Temperley–Lieb algebra may be generic, yielding a quantum version of the geometric Satake equivalence for $\mathfrak{sl}_{2}$. When $W$ is finite, $q$ must be specialized to an appropriate root of unity, and the negligible Jones–Wenzl projector yields the Soergel bimodule for the longest element of $W$.