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We introduce a family of generalized Schr\"oder polynomials $S_\tau(q,t,a)$, indexed by triangular partitions $\tau$ and prove that $S_\tau(q,t,a)$ agrees with the Poincar\'e series of the triply graded Khovanov-Rozansky homology of the Coxeter knot $K_\tau$ associated to $\tau$. For all integers $m,n,d\geq 1$ with $m,n$ relatively prime, the $(d,mnd+1)$-cable of the torus knot $T(m,n)$ appears as a special case. It is known that these knots are algebraic, and as a result we obtain a proof of the $q=1$ specialization of the Oblomkov-Rasmussen-Shende conjecture for these knots. Finally, we show that our Schr\"oder polynomial computes the hook components in the Schur expansion of the symmetric function appearing in the shuffle theorem under any line, thus proving a triangular version of the $(q,t)$-Schr\"oder theorem., Comment: 53 pages, 17 figures

To every compact oriented surface that is composed entirely out of 2-dimensional 0- and 1-handles, we construct a dg category using structures arising in Khovanov homology. These dg categories form part of the 2-dimensional layer (a.k.a. modular functor) of a categorified version of the sl(2) Turaev--Viro topological field theory. As a byproduct, we obtain a unified perspective on several hitherto disparate constructions in categorified quantum topology, including the Rozansky--Willis invariants, Asaeda--Przytycki--Sikora homologies for links in thickened surfaces, categorified Jones--Wenzl projectors and associated spin networks, and dg horizontal traces., Comment: 58 pages, best viewed in color, comments welcome

We show that the Khovanov and Cooper-Krushkal models for colored sl(2) homology are equivalent in the case of the unknot, when formulated in the quantum annular Bar-Natan category. Again for the unknot, these two theories are shown to be equivalent to a third colored homology theory, defined using the action of Jones-Wenzl projectors on the quantum annular homology of cables. The proof is given by conceptualizing the properties of all three models into a Chebyshev system and by proving its uniqueness. In addition, we show that the classes of the Cooper-Hogancamp projectors in the quantum horizontal trace coincide with those of the Cooper-Krushkal projectors on the passing through strands. As an application, we compute the full quantum Hochschild homology of Khovanov's arc algebras. Finally, we state precise conjectures formalizing cabling operations and extending the above results to all knots., Comment: 47 pages, color figures (but can be safely printed black and white)

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., Comment: 47 pages, best viewed in color, comments welcome. v2 clarifies the abstract and introduction

We introduce a multi-parameter deformation of the triply-graded Khovanov--Rozansky homology of links colored by one-column Young diagrams, generalizing the "$y$-ified" link homology of Gorsky--Hogancamp and work of Cautis--Lauda--Sussan. For each link component, the natural set of deformation parameters corresponds to interpolation coordinates on the Hilbert scheme of the plane. We extend our deformed link homology theory to braids by introducing a monoidal dg 2-category of curved complexes of type $A$ singular Soergel bimodules. Using this framework, we promote to the curved setting the categorical colored skein relation from arXiv:2107.08117 and also the notion of splitting map for the colored full twists on two strands. As applications, we compute the invariants of colored Hopf links in terms of ideals generated by Haiman determinants and use these results to establish general link splitting properties for our deformed, colored, triply-graded link homology. Informed by this, we formulate several conjectures that have implications for the relation between (colored) Khovanov--Rozansky homology and Hilbert schemes., Comment: 110 pages, many figures, color viewing helpful

We study the skein relation that governs the HOMFLYPT invariant of links colored by one-column Young diagrams. Our main result is a categorification of this colored skein relation. This takes the form of a homotopy equivalence between two one-sided twisted complexes constructed from Rickard complexes of singular Soergel bimodules associated to braided webs. Along the way, we prove a conjecture of Beliakova--Habiro relating the colored 2-strand full twist complex with the categorical ribbon element for quantum $\mathfrak{sl}_2$., Comment: 34 pages, many diagrams

We define a new family of commuting operators $F_k$ in Khovanov-Rozansky link homology, similar to the action of tautological classes in cohomology of character varieties. We prove that $F_2$ satisfies ``hard Lefshetz property" and hence exhibits the symmetry in Khovanov-Rozansky homology conjectured by Dunfield, Gukov and Rasmussen., Comment: 64 pages, accepted in Duke Math. J

We give a general construction of categorical idempotents which recovers the categorified Jones-Wenzl projectors, categorified Young symmetrizers, and other constructions as special cases. The construction is intimately tied to cell theory in the sense of additive monoidal categories., Comment: 25 pages; v2 fixes typos and some missing figures

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. Secondly, we introduce the notion of derived horizontal trace of a monoidal dg category and compute the derived horizontal trace of Soergel bimodules in type A. 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., Comment: 74 pages, comments welcome

We prove a general version of the homological perturbation lemma which works in the presence of curvature, and without the restriction to strong deformation retracts, building on work of Markl. A key observation is that the notion of strong homotopy equivalence of complexes (or objects in an abstract dg category) has a natural expression in the language of curved twisted complexes., Comment: 21 pages. v2 adds section 2.5 on twists of contractible complexes

We compute the triply graded Khovanov-Rozansky homology of a family of links, including positive torus links and $\operatorname{Sym}^l$-colored torus knots., Comment: Many figures

We construct an explicit equivalence between the (bi)category of gl(2) webs and foams and the Bar-Natan (bi)category of Temperley-Lieb diagrams and cobordisms. With this equivalence we can fix functoriality of every link homology theory that factors through the Bar-Natan category. To achieve this, we define web versions of arc algebras and their quasi-hereditary covers, which provide strictly functorial tangle homologies. Furthermore, we construct explicit isomorphisms between these algebras and the original ones based on Temperley-Lieb cup diagrams. The immediate application is a strictly functorial version of the Beliakova-Putyra-Wehrli quantization of the annular link homology., Comment: 44 pages, color pictures (but printing in black and white is OK)

The Hecke category participates in an equivalence called monoidal Koszul duality, which exchanges it with the category of (Langlands-dual) "free-monodromic tilting sheaves." Motivated by a recent conjecture of Gorsky and the first-named author on HOMFLYPT link homology, we propose to enhance this duality with an additional grading. We provide evidence for this enhancement in the case of $\mathrm{GL}_2$, working in the language of the second-named author's joint work with Achar, Riche, and Williamson., Comment: 17 pages; minor edits to address referee's comments; final version

We prove that the full twist is a Serre functor in the homotopy category of type A Soergel bimodules. As a consequence, we relate the top and bottom Hochschild degrees in Khovanov-Rozansky homology, categorifying a theorem of K\'alm\'an., Comment: 24 pages

For a diagram of a 2-stranded tangle in the 3-ball we define a twisted complex of compact Lagrangians in the triangulated envelope of the Fukaya category of the smooth locus of the pillowcase. We show that this twisted complex is a functorial invariant of the isotopy class of the tangle, and that it provides a factorization of Bar-Natan's functor from the tangle cobordism category to chain complexes. In particular, the hom set of our invariant with a particular non-compact Lagrangian associated to the trivial tangle is naturally isomorphic to the reduced Khovanov chain complex of the closure of the tangle. Our construction comes from the geometry of traceless SU(2) character varieties associated to resolutions of the tangle diagram, and was inspired by Kronheimer and Mrowka's singular instanton link homology., Comment: 53 pages, 16 figures; Improvements to exposition. This is essentially the published version

We conjecture that the complex of Soergel bimodules associated with the full twist braid is categorically diagonalizable, for any finite Coxeter group. This utilizes the theory of categorical diagonalization introduced earlier by the authors. We prove our conjecture in type $A$, and as a result we obtain a categorification of the Young idempotents.

We define a deformation of the triply graded Khovanov-Rozansky homology of a link $L$ depending on a choice of parameters $y_c$ for each component of $L$, which satisfies link-splitting properties similar to the Batson-Seed invariant. Keeping the $y_c$ as formal variables yields a link homology valued in triply graded modules over $\mathbb{Q}[x_c,y_c]_{c\in \pi_0(L)}$. We conjecture that this invariant restores the missing $Q\leftrightarrow TQ^{-1}$ symmetry of the triply graded Khovanov-Rozansky homology, and in addition satisfies a number of predictions coming from a conjectural connection with Hilbert schemes of points in the plane. We compute this invariant for all positive powers of the full twist and match it to the family of ideals appearing in Haiman's description of the isospectral Hilbert scheme., Comment: 61 pages

Using the theory of involutive Heegaard Floer knot theory developed by Hendricks-Manolescu, we define two involutive analogs of the Upsilon knot concordance invariant of Ozsvath-Stipsicz-Szabo. These involutive invariants are piecewise linear functions defined on the interval [0,2]. Each is a concordance invariant and provides bounds on the three-genus of a knot., Comment: 9 pages

This paper lays the groundwork for the theory of categorical diagonalization. Given a diagonalizable operator, tools in linear algebra (such as Lagrange interpolation) allow one to construct a collection of idempotents which project to each eigenspace. These idempotents are mutually orthogonal and sum to the identity. We categorify these tools. At the categorical level, one has not only eigenobjects and eigenvalues but also eigenmaps, which relate an endofunctor to its eigenvalues. Given an invertible endofunctor of a triangulated category with a sufficiently nice collection of eigenmaps, we construct idempotent functors which project to eigencategories. These idempotent functors are mutually orthogonal, and a convolution thereof is isomorphic to the identity functor. In several sequels to this paper, we will use this technology to study the categorical representation theory of Hecke algebras. In particular, for Hecke algebras of type A, we will construct categorified Young symmetrizers by simultaneously diagonalizing certain functors associated to the full twist braids.

We give a simple recursion which computes the triply graded Khovanov-Rozansky homology of several infinite families of knots and links, including the $(n,nm\pm 1)$ and $(n,nm)$ torus links for $n,m\geq 1$. We interpret our results in terms of Catalan combinatorics, proving a conjecture of Gorsky's. Our computations agree with predictions coming from Hilbert schemes and rational DAHA, which also proves the Gorsky-Oblomkov-Rasmussen-Shende conjectures in these cases. Additionally, our results suggest a topological interpretation of the symmetric functions which appear in the context of the $m$-shuffle conjecture of Haglund-Haiman-Loehr-Remmel-Ulyanov., Comment: 39 pages, many figures

We investigate analogs of symmetric functions arising from an extension of the nilHecke algebra defined by Naisse and Vaz. These extended symmetric functions form a subalgebra of the polynomial ring tensored with an exterior algebra. We define families of bases for this algebra and show that it admits a family of differentials making it a sub-DG-algebra of the extended nilHecke algebra. The ring of extended symmetric functions equipped with this differential is quasi-isomorphic to the cohomology of a Grassmannian. We also introduce new deformed differentials on the extended nilHecke algebra that when restricted makes extended symmetric functions quasi-isomorphic to $GL(N)$-equivariant cohomology of Grassmannians., Comment: v2. Minor corrections, reference updated. Material added: connection between extended symmetric functions and the work of Solomon (Section 4). 30 pages

In these notes we develop some basic theory of idempotents in monoidal categories. We introduce and study the notion of a pair of complementary idempotents in a triangulated monoidal category, as well as more general idempotent decompositions of identity. If $\mathbf{E}$ is a categorical idempotent then $\operatorname{End}(\mathbf{E})$ is a graded commutative algebra. The same is true of $\operatorname{Hom}(\mathbf{E},\mathbf{E}^c[1])$ under certain circumstances, where $\mathbf{E}^c$ is the complement. These generalize the notions of cohomology and Tate cohomology of a finite dimensional Hopf algebra, respectively.

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 Gorsky et al relating homology of torus links with Hilbert schemes.

We construct complexes $P_{1^n}$ of Soergel bimodules which categorify the Young idempotents corresponding to one-column partitions. A beautiful recent conjecture of Gorsky-Rasmussen relates the Hochschild homology of categorified Young idempotents with the flag Hilbert scheme. We prove this conjecture for $P_{1^n}$ and its twisted variants. We also show that this homology is also a certain limit of Khovanov-Rozansky homologies of torus links. Along the way we obtain several combinatorial results which could be of independent interest., Comment: 57 pages, many TikZ figures

We show that the triply graded Khovanov-Rozansky homology of the torus link $T_{n,k}$ stablizes as $k\to \infty$. We explicitly compute the stable homology (as a ring), which proves a conjecture of Gorsky-Oblomkov-Rasmussen-Shende. To accomplish this, we construct complexes $P_n$ of Soergel bimodules which categorify the Young symmetrizers corresponding to one-row partitions and show that $P_n$ is a stable limit of Rouquier complexes. A certain derived endomorphism ring of $P_n$ computes the aforementioned stable homology of torus links., Comment: Significant changes from v1. Changed title. Restructured to improve readability. The appendix on categorical idempotents has been removed and posted separately as arXiv:1703.01001