Motivated by topology, we develop a general theory of traces and shadows for an endobicategory, which is a pair: bicategory C and endobifunctor Σ : C C. For a graded linear bicategory and a fixed invertible parameter q, we quantize this theory by using the endofunctor Σ q such that Σ q α := q − deg α Σα for any 2-morphism α and coincides with Σ otherwise.Applying the quantized trace to the bicategory of Chen-Khovanov bimodules we get a new triply graded link homology theory called quantum annular link homology. If q = 1 we reproduce Asaeda-Przytycki-Sikora (APS) homology for links in a thickened annulus. We prove that our homology carries an action of U q (sl 2 ), which intertwines the action of cobordisms. In particular, the quantum annular homology of an n-cable admits an action of the braid group, which commutes with the quantum group action and factors through the Jones skein relation. This produces a nontrivial invariant for surfaces knotted in four dimensions. Moreover, a direct computation for torus links shows that the rank of quantum annular homology groups does depend on the quantum parameter q.

Abstract. We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further, we study conditions under which framed colored link cobordisms induce chain transformations between our formal brackets. We conjecture that for special choice of parameters, Khovanov and Lee homology theories of colored links are functorial (up to sign). Finally, we extend the Rasmussen invariant to links and give examples where this invariant is a stronger obstruction to sliceness than the multivariable Levine-Tristram signature.

Chronic metabolic acidosis increases renal cortical NaDC-1 mRNA abundance and apical membrane NaDC-1 protein abundance, while alkali feeding is without effect on NaDC-1.

Let L ⊂ A × I be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of sl 2 (∧), the exterior current algebra of sl 2 . When L is an m-framed n-cable of a knot K ⊂ S 3 , its sutured annular Khovanov homology carries a commuting action of the symmetric group S n . One therefore obtains a 'knotted' Schur-Weyl representation that agrees with classical sl 2 Schur-Weyl duality when K is the Seifert-framed unknot.

We show that the Khovanov complex of a connected link diagram D retracts to a subcomplex whose generators are in 2 : 1 correspondence with the spanning trees of the "black graph" of D. Using this result, we give a new proof of Lee's theorem on the support of Khovanov homology of alternating knots.