  We prove an excision theorem for the singular instanton Floer homology that
allows the excision surfaces to intersect the singular locus. This is an
extension of the non-singular excision theorem by Kronheimer and Mrowka and the
genus-zero singular excision theorem by Street. We use the singular excision
theorem to define an instanton Floer homology theory for sutured manifolds with
tangles. As applications, we prove that the annular Khovanov homology (1)
detects the unlink, (2) detects the closure of the trivial braid, and (3)
distinguishes braid closures from other links; we also prove that the annular
instanton Floer homology detects the Thurston norm of meridional surfaces.
