  We study the exotic t-structure on the derived category of coherent sheaves
on two-block Springer fibre (i.e. for a nilpotent matrix of type (m+n,n) in
type A). The exotic t-structure has been defined by Bezrukavnikov and Mirkovic
for Springer theoretic varieties in order to study representations of Lie
algebras in positive characteristic. Using work of Cautis and Kamnitzer, we
construct functors indexed by affine tangles, between categories of coherent
sheaves on different two-block Springer fibres (i.e. for different values of
n). After checking some exactness properties of these functors, we describe the
irreducible objects in the heart of the exotic t-structure, and enumerate them
by crossingless (m,m+2n) matchings. We compute the Ext's between the
irreducible objects, and show that the resulting algebras are an annular
variant of Khovanov's arc algebras. In subsequent work we will make a link with
annular Khovanov homology, and use these results to give a positive
characteristic analogue of some categorification results using two-block
parabolic category O (by Bernstein-Frenkel-Khovanov, Brundan, Stroppel, et al).
