The main result of this paper asserts that the convolution algebra associated with a \(2\)-block Springer fiber is isomorphic to a certain combinatorially described algebra with basis given by certain arc diagrams (and inspired by diagrammatic description of Khovanov homology). A similar result holds for the extended version of the convolution algebra and the quasi-hereditary cover of the generalized arc algebra. The generalized arc algebra also describes perverse sheaves on a Grassmanian and endomorphims of a basic projective-injective module in a maximal parabolic block of the BGG category \(\mathcal{O}\). The same algebra has several other geometric incarnations which are also investigated.