On the Categorified Wrapping Number Conjecture

We prove the Categorified Wrapping Number Conjecture for large classes of annular links, including alternating annular links and tangle closures exhibiting plumbed link phenomena. We do so by characterizing when a resolution is sufficient to produce a nonzero homology class in k-grading wrap(L) on its own. This characterization primarily concerns the type of crossing resolutions abutting trivial circles. We also discuss generalizations of the Conjecture to a thickened g-punctured disk, and extend our results to this setting.