Graded Link Homology Research Articles

Categorical Chern character and braid groups

To a braid β∈Brn we associate a complex of sheaves Sβ on Hilbn(C2) such that the previously defined triply graded link homology of the closure L(β) is isomorphic to the homology of Sβ. The construction of Sβ relies on the Chern functor CH:MFnst→DC⁎×C⁎per(Hilbn(C2)) defined in the paper together with its adjoint functor HC.

A quantum categorification of the Alexander polynomial

Using a modified foam evaluation, we give a categorification of the Alexander polynomial of a knot. We also give a purely algebraic version of this knot homology which makes it appear as the infinite page of a spectral sequence starting at the reduced triply graded link homology of Khovanov--Rozansky.

2-Verma modules and the Khovanov–Rozansky link homologies

We explain how Queffelec–Sartori’s construction of the HOMFLY-PT link polynomial can be interpreted in terms of parabolic Verma modules for \({\mathfrak {gl}_{2n}}\). Lifting the construction to the world of categorification, we use parabolic 2-Verma modules to give a higher representation theory construction of Khovanov–Rozansky’s triply graded link homology.

Symmetric Khovanov-Rozansky link homologies

On donne une catégorification de l’évaluation symétrique des toiles 𝔰𝔩 N en utilisant les mousses. On en déduit des théories homologiques d’entrelacs qui catégorifient les invariants quantiques d’entrelacs associés aux puissances symétriques de la représentation standard de 𝔰𝔩 N . Ces théories sont obtenues dans un cadre équivariant. On montre qu’il existe des suites spectrales de l’homologie triplement graduée de Khovanov-Rozansky vers ces homologies symétriques. On donne aussi une interpretation des bimodules de Soergel en terme de mousses.

Quantum link homology via trace functor I

Motivated by topology, we develop a general theory of traces and shadows for an endobicategory, which is a pair: bicategory and endobifunctor . For a graded linear bicategory and a fixed invertible parameter q, we quantize this theory by using the endofunctor $$\Sigma _q$$ such that $$\Sigma _q \alpha :=q^{-\deg \alpha }\Sigma \alpha $$ for any 2-morphism $$\alpha $$ and coincides with $$\Sigma $$ 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 homology for links in a thickened annulus. We prove that our homology carries an action of , 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 depend on the quantum parameter q.

On the computation of torus link homology

We introduce a new method for computing triply graded link homology, which is particularly well adapted to torus links. Our main application is to the$(n,n)$-torus links, for which we give an exact answer for all$n$. In several cases, our computations verify conjectures of Gorskyet al.relating homology of torus links with Hilbert schemes.

Virtual crossings and a filtration of the triply graded link homology of a link diagram

A filtration of Soergel bimodules by virtual crossing bimodules extends to Rouquier’s complexes associated with braid words. We show that these complexes are invariant up to filtered homotopy with respect to the second Reidemeister move, and the filtration of the triply graded link diagram homology, constructed by Khovanov through the application of the Hochschild homology, is invariant under Markov moves. We also prove that the homotopy equivalence of the complexes of braid words related by the third Reidemeister move violates filtration by at most two units.

Positive half of the Witt algebra acts on triply graded link homology

The positive half of the Witt algebra is the Lie algebra spanned by vector fields x^{m+1} \frac {d}{dx} acting as differentiations on the polynomial algebra \mathbb Q[x] upon which the Soergel bimodule construction of triply graded link homology is based. We show that this action of Witt algebra can be extended to the link homology.

Bar-Natan’s geometric complex and a categorification of the Dye–Kauffman–Miyazawa polynomial

In this paper, we construct a categorification of the two-variable Dye–Kauffman–Miyazawa polynomial by utilizing Bar-Natan’s construction of the Khovanov homology and homotopy quantum field theories (HQFTs) given by Turaev. In particular, for any stable equivalence class, we construct a [Formula: see text]-graded link homology over [Formula: see text] whose graded Euler characteristic is the two-variable Dye–Kauffman–Miyazawa polynomial. Moreover, we show that it is isomorphic to a special case of Dye–Kauffman–Manturov’s categorification. In this sense, we explain the special case of Dye–Kauffman–Manturov’s homology in terms of Bar-Natan’s construction.

The 1,2-coloured HOMFLY-PT link homology

In this paper we define the 1,2-coloured HOMFLY-PT triply graded link homology and prove that it is a link invariant. We also conjecture on how to generalize our construction for arbitrary colours.

A geometric model for Hochschild homology of Soergel bimodules

An important step in the calculation of the triply graded link homology theory of Khovanov and Rozansky is the determination of the Hochschild homology of Soergel bimodules for SL(n). We present a geometric model for this Hochschild homology for any simple group G, as equivariant intersection homology of B x B-orbit closures in G. We show that, in type A these orbit closures are equivariantly formal for the conjugation T-action. We use this fact to show that in the case where the corresponding orbit closure is smooth, this Hochschild homology is an exterior algebra over a polynomial ring on generators whose degree is explicitly determined by the geometry of the orbit closure, and describe its Hilbert series, proving a conjecture of Jacob Rasmussen.