  We compute symplectic cohomology for Milnor fibres of certain compound Du Val
singularities that admit small resolution by using homological mirror symmetry.
Our computations suggest a new conjecture that the existence of a small
resolution has strong implications for the symplectic cohomology and
conversely. We also use our computations to give a contact invariant of the
link of the singularities and thereby distinguish many contact structures on
connected sums of $S^2 \times S^3$.
