Relations in singular instanton homology

We calculate the singular instanton homology with local coefficients for the simplest n-strand braids in $S^1 \times S^2$ for all odd n, describing these homology groups and their module structures in terms of the coordinate rings of explicit algebraic curves. The calculation is expected to be equivalent to computing the quantum cohomology ring of a certain Fano variety, namely a moduli space of stable parabolic bundles on a sphere with n marked points.