Using Turaev’s observation that the HOMFLYPT skein of the annulus is the ring of q-symmetric functions, we can expand the annular closure of any braid in any basis for the latter. For positive braid closures, I gave a geometric interpretation of the modified Hall–Littlewood expansion in terms of braid varieties with unipotent monodromy. I will explain how this yields an equivalence between a 1975 point-counting identity of Kawanaka and a 2009 theorem of Kálmán about HOMFLYPT, which turns out to lift to an equivalence of respective categorifications. In more recent work, Nathan Williams and I interpret Kawanaka's most general identity in terms of partial Springer resolutions. I will explain how this allows us to decompose Springer-theoretic braid varieties into pieces with the point counts, though not the geometry, of Deodhar cells. If time permits, I will show how these results predict surprising homotopy equivalences between varieties over the complex numbers.