Paul Wedrich

We construct a Kirby color in the setting of Khovanov homology: an ind-object of the annular Bar-Natan category that is equipped with a natural handle slide isomorphism. Using functoriality and cabling properties of Khovanov homology, we define a Kirby-colored Khovanov homology that is invariant under the handle slide Kirby move, up to isomorphism. Via the Manolescu–Neithalath 2-handle formula, Kirby-colored Khovanov homology agrees with the \mathfrak{gl}_{2} skein lasagna module, hence is an invariant of 4-dimensional 2-handlebodies.

We prove that the generating functions for the one row/column colored HOMFLY-PT invariants of arborescent links are specializations of the generating functions of the motivic Donaldson-Thomas invariants of appropriate quivers that we naturally associate with these links. Our approach extends the previously established tangles-quivers correspondence for rational tangles to algebraic tangles by developing gluing formulas for HOMFLY-PT skein generating functions under Conway's tangle addition. As a consequence, we prove the conjectural links-quivers correspondence of Kucharski-Reineke-Sto\v{s}i\'c-Sulkowski for all arborescent links.

Using diagrammatic methods, we define a quiver with relations depending on a prime p \mathsf {p} and show that the associated path algebra describes the category of tilting modules for S L 2 \mathrm {SL}_{2} in characteristic p \mathsf {p} . Along the way we obtain a presentation for morphisms between p \mathsf {p} -Jones–Wenzl projectors.

Using diagrammatic methods, we define a quiver algebra depending on a prime p and show that it is the algebra underlying the category of tilting modules for SL(2) in characteristic p. Along the way we obtain a presentation for morphisms between p-Jones-Wenzl projectors.

For every oriented surface of finite type, we construct a functorial Khovanov homology for links in a thickening of the surface, which takes values in a categorification of the corresponding \mathfrak {gl}_2 skein module. The latter is a mild refinement of the Kauffman bracket skein algebra, and its categorification is constructed using a category of \mathfrak {gl}_2 foams that admits an interesting non-negative grading. We expect that the natural algebra structure on the \mathfrak {gl}_2 skein module can be categorified by a tensor product that makes the surface link homology functor monoidal. We construct a candidate bifunctor on the target category and conjecture that it extends to a monoidal structure. This would give rise to a canonical basis of the associated \mathfrak {gl}_2 skein algebra and verify an analogue of a positivity conjecture of Fock and Goncharov and Thurston. We provide evidence towards the monoidality conjecture by checking several instances of a categorified Frohman–Gelca formula for the skein algebra of the torus. Finally, we recover a variant of the Asaeda–Przytycki–Sikora surface link homologies and prove that surface embeddings give rise to spectral sequences between them.

In this note, we compute the centers of the categories of tilting modules for G = SL₂ in prime characteristic, of tilting modules for the corresponding quantum group at a complex root of unity, and of projective G_gT-modules when g = 1, 2.

In this note we compute the centers of the categories of tilting modules for G=SL2 in prime characteristic, of tilting modules for the corresponding quantum group at a complex root of unity, and of projective GgT-modules when g=1,2.