Embedded surface invariants via the Broda-Petit construction

We recall Petit’s construction of “dichromatic” invariants of 4-manifolds computed from Kirby diagrams using a nested pair of ribbon fusion categories [Formula: see text] as initial data. Along the way we prove a lemma that fits the use of formal linear combinations of simple objects with quantum dimensions a coefficients as in the constructions of Reshetikhin–Turaev, Broda, and Petit more firmly in the functorial framework favored by the authors. We then show that Hughes et al.’s banded-link presentations of surfaces embedded in 4-manifolds provide a means whereby Frobenius algebra in [Formula: see text] together with a suitable module over it lying in [Formula: see text] give rise to an invariant of a surface-4-manifold pair. Observing that the definition of the invariants nowhere uses the slice disks needed for banded unlinks to encode surfaces, we extend the construction to Kirby diagrams equipped with links, modulo the same combinatorial moves. We provide a class of examples of suitable initial data and compute sufficient examples to show the invariant is sensitive to genus and knotting, and via an example of Akbulut, that the extension is sensitive to smooth structure.