15) Derived traces of Soergel categories
We study two kinds of categorical traces of (monoidal) dg
categories, with particular interest in categories of
Soergel bimodules. First, we explicitly compute the
usual Hochschild homology, or derived vertical trace, of
the category of Soergel bimodules in arbitrary types.
We show that this dg algebra is formal, and calculate
its homology explicitly, for all Coxeter groups.
Secondly, we introduce the notion of derived horizontal
trace of a monoidal dg category and identify the derived
horizontal trace of Soergel bimodules in type $A$ with
the homotopy category of perfect dg modules of an
explicit algebra. As an application we obtain a derived
annular Khovanov--Rozansky link invariant with an action
of full twist insertion, and thus a categorification of
the HOMFLY-PT skein module of the solid torus.
With Eugene Gorsky and Matthew Hogancamp
14) Invariants of 4-manifolds from Khovanov-Rozansky link homology
We use Khovanov-Rozansky gl(N) link homology to define invariants of oriented smooth 4-manifolds, as skein modules constructed from certain 4-categories with well-behaved duals. The technical heart of this construction is a proof of the sweep-around property, which makes these link homologies well defined in the 3-sphere.
With Scott Morrison and Kevin Walker
13) Quivers for SL(2) tilting modules
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
With Daniel Tubbenhauer
12) Evaluations of annular Khovanov-Rozansky homology
We describe the universal target of annular Khovanov-Rozansky link homology functors as
the homotopy category of a free symmetric monoidal category generated by one object and
one endomorphism. This categorifies the ring of symmetric functions and admits
categorical analogues of plethystic transformations, which we use to characterize the
annular invariants of Coxeter braids. Further, we prove the existence of symmetric group
actions on the Khovanov-Rozansky invariants of cabled tangles and we introduce spectral
sequences that aid in computing the homologies of generalized Hopf links. Finally, we
conjecture a characterization of the horizontal traces of Rouquier complexes of Coxeter
braids in other types.
With Eugene Gorsky.
11) Khovanov homology and categorification of skein modules
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 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 gl(2)
foams that admits an interesting non-negative grading. We expect that the natural
algebra structure on the 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 gl(2) skein algebra and verify an
analogue of a positivity conjecture of Fock–Goncharov and Turston. 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.
10) Extremal weight projectors II
In previous work, we have constructed diagrammatic idempotents in an affine extension of
the Temperley–Lieb category, which describe extremal weight projectors for sl(2), and
which categorify Chebyshev polynomials of the first kind. In this paper, we generalize
the construction of extremal weight projectors to the case of gl(N) for N > 1, with a
view towards categorifying the corresponding torus skein algebras via Khovanov–Rozansky
link homology. As by-products, we obtain compatible diagrammatic presentations of the
representation categories of gl(N) and its Cartan subalgebra, and a categorification of
power-sum symmetric polynomials.
9) Algèbre diagrammatique et catégorification
Nous proposons une illustration diagrammatique
abordable du concept de catégorification qui s’est
développé au cours des vingt dernières années.
A friendly outreach article in French, written mostly by Hoel
Queffelec, inspired by our joint work.
La Gazette des mathématiciens 163 (janvier 2020)
8) Rational links and DT invariants of quivers
We prove that the generating functions for the colored HOMFLY-PT polynomials of rational
links are specializations of the generating functions of the motivic Donaldson-Thomas
invariants of appropriate quivers that we naturally associate with these links. This
shows that the conjectural links-quivers correspondence of
Kucharski–Reineke–Stošić–Sułkowski as well as the LMOV conjecture hold
for rational links. Along the way, we extend the links-quivers correspondence to tangles
and, thus, explore elements of a skein theory for motivic Donaldson-Thomas invariants.
With Marko Stošić.
International Mathematical Research Notices
7) Functoriality of colored link homologies
We prove that the bigraded, colored Khovanov–Rozansky type A link and tangle invariants
are functorial with respect to link and tangle cobordisms.
With Michael Ehrig
and Daniel Tubbenhauer.
Proceedings of the London Mathematical Society 117-5 (2018), 996-1040
6) Extremal weight projectors
We introduce a quotient of the affine Temperley-Lieb category that encodes all weight
preserving linear maps between finite-dimensional sl(2)-representations. We study the
diagrammatic idempotents that correspond to projections onto extremal weight spaces and
find that they satisfy similar properties as Jones-Wenzl projectors, and that they
categorify the Chebyshev polynomials of the first kind. This gives a categorification of
the Kauffman bracket skein algebra of the annulus, which is well adapted to the task of
categorifying the multiplication on the Kauffman bracket skein module of the torus.
Mathematical Research Letters 25-6 (2018), 1911-1936
5) Exponential growth of colored HOMFLY-PT homology
We define reduced colored sl(N) link homologies and use deformation spectral sequences to
characterize their dependence on color and rank. We then define reduced colored
HOMFLY-PT homologies and prove that they arise as large N limits of sl(N) homologies.
Together, these results allow proofs of many aspects of the physically conjectured
structure of the family of type A link homologies. In particular, we verify a conjecture
of Gorsky, Gukov and Stošić about the growth of colored HOMFLY-PT
Advances in Mathematics 353 (2019), 471-525
4) Super q-Howe duality and web categories
We use super q–Howe duality to provide diagrammatic presentations of an idempotented form
of the Hecke algebra and of categories of gl(N)–modules (and, more generally,
gl(N|M)–modules) whose objects are tensor generated by exterior and symmetric powers of
the vector representations. As an application, we give a representation-theoretic
explanation and a diagrammatic version of a known symmetry of colored HOMFLY–PT
With Daniel Tubbenhauer and Pedro Vaz.
Algebraic & Geometric Topology 17-6 (2017), 3703-3749
3) Deformations of colored sl(N) link homologies via foams
We prove a conjectured decomposition of deformed sl(N) link homology, as well as an
extension to the case of colored links, generalizing results of Lee, Gornik, and Wu. To
this end, we use foam technology to give a completely combinatorial construction of Wu’s
deformed colored sl(N) link homologies. By studying the underlying deformed higher
representation-theoretic structures and generalizing the Karoubi envelope approach of
Bar-Natan and Morrison, we explicitly compute the deformed invariants in terms of
undeformed type A link homologies of lower rank and color.
With David Rose.
Geometry & Topology 20-6 (2016), 3431-3517
2) q-holonomic formulas for colored HOMFLY polynomials of 2-bridge links
We compute q-holonomic formulas for the HOMFLY polynomials of 2-bridge links colored with
one-column (or one-row) Young diagrams.
Journal of Pure and Applied Algebra 223-4 (2019), 1434-1439
1) Categorified sl(N) invariants of colored rational tangles
We use categorical skew Howe duality to find recursion rules that compute categorified
sl(N) invariants of rational tangles colored by exterior powers of the standard
representation. Further, we offer a geometric interpretation of these rules which
suggests a connection to Floer theory. Along the way we make progress towards two
conjectures about the colored HOMFLY homology of rational links and discuss consequences
for the corresponding decategorified invariants.
Algebraic & Geometric Topology 16-1 (2016), 427-482