Instanton Floer homology for sutured manifolds with tangles

Referee: [Excision theorem (main statement and proof)] The central excision theorem requires a local model and index computation for the deformation operator at points where the excision surface meets the singular locus. The manuscript must supply an explicit transversality or surjectivity argument for this new case (distinct from the genus-zero analysis of Street) to justify the isomorphism; without it the subsequent definition of the sutured-tangle homology and all applications rest on an unverified analytic step.

Authors: Section 3.2 of the manuscript supplies the local model for the deformation operator at excision-surface/singular-locus intersection points. Proposition 4.2 then computes the index and establishes surjectivity via an explicit Fredholm analysis that extends Street's genus-zero case by incorporating the additional elliptic terms arising from positive-genus surfaces and tangle intersections; the argument relies on the same weighted Sobolev spaces and perturbation scheme used in the non-singular Kronheimer-Mrowka setting. We therefore maintain that the isomorphism is rigorously justified, though we can expand the transversality paragraph for added clarity. revision: partial

Referee: [Applications section] The applications to annular Khovanov homology detection (unlink, trivial braid, braid vs. non-braid distinction) and to Thurston-norm detection are derived from the new homology groups. These claims are load-bearing only if the excision isomorphism is rigorously established; the current presentation leaves the analytic foundation for the isomorphism as a potential gap.

Authors: The applications in Section 5 are direct consequences of the excision isomorphism proved in Theorem 1.1 and the subsequent definition of the sutured-tangle homology. The analytic steps are already referenced in the proof of the main theorem; we can insert additional cross-references in Section 5 to make the dependence explicit. revision: partial