Curves in the disc, the type B braid group, and the type B zigzag algebra

We construct a finite dimensional quiver algebra from the non-simply laced type $B$ Dynkin diagram, which we call the type $B$ zigzag algebra. This leads to a faithful categorical action of the type $B$ braid group $\mathcal{A}(B)$, acting on the homotopy category of its projective modules. This categorical action is also closely related to the topological action of $\mathcal{A}(B)$, viewed as mapping class group of the punctured disc -- hence our exposition can be seen as a type $B$ analogue of Khovanov-Seidel's work in arXiv:math/0006056v2. Moreover, we show that certain category of bimodules over our type $B$ zigzag algebra is a quotient category of Soergel bimodules, resulting in an alternative proof to Rouquier's conjecture on the faithfulness of the 2-braid groups for type $B$.