Weak representation of tangle categories in algebraic geometry

In this paper we develop a formalism of spherical functors and apply it to construct weak representations of the category of affine tangles up to isotopies. We prove that under certain natural conditions these weak representations give rise to strong representations of an enveloping category (the category of affine tangles up to flat isotopies). We apply these ideas to study a particular representation of the category of framed affine tangles in derived categories of coherent sheaves on Springer fibers. We obtain a diagrammatical description of irreducible objects in the heart of the exotic t-structures on those derived categories, and describe the algebra of endomorphisms of the direct sum of those irreducibles.