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We prove an excision theorem for the singular instanton Floer homology thatallows the excision surfaces to intersect the singular locus. This is anextension of the non-singular excision theorem by Kronheimer and Mrowka and thegenus-zero singular excision theorem by Street. We use the singular excisiontheorem to define an instanton Floer homology theory for sutured manifolds withtangles. 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 annularinstanton Floer homology detects the Thurston norm of meridional surfaces.

Instanton Floer homology for sutured manifolds with tangles