  Sutured instanton Floer homology was introduced by Kronheimer and Mrowka. In
this paper, we prove that for a taut balanced sutured manifold with vanishing
second homology, the dimension of the sutured instanton Floer homology provides
a bound on the minimal depth of all possible taut foliations on that balanced
sutured manifold. The same argument can be adapted to the monopole and even the
Heegaard Floer settings, which gives a partial answer to one of Juhasz's
conjectures. Using the nature of instanton Floer homology, on knot complements,
we can construct a taut foliation with bounded depth, given some information on
the representation varieties of the knot fundamental groups. This indicates a
mystery relation between the representation varieties and some small depth taut
foliations on knot complements, and gives a partial answer to one of Kronheimer
and Mrowka's conjecture.
