We will study the category of modular representations of the special linear Lie algebra with a fixed two-block nilpotent p-character, by using geometric techniques of Bezrukavnikov, Mirkovic and Rumynin that express it in terms of coherent sheaves on the corresponding Springer fiber. Building on work of Cautis and Kamnitzer, we construct a categorification of the affine tangle calculus using these categories, and describe the coherent sheaves that corresponds to the irreducible representations. Using this, we give character and dimension formulae, descriptions of the Ext spaces, and a (conjectural) description of this category as modules over an affinization of Khovanov's arc algebra.