arXiv subjects

Matthew Hogancamp

Publications and source records attributed to Matthew Hogancamp.

At least 19 recordsLinked to original sources

arXiv2025

Stable deformed $\mathfrak{gl}_N$ homology of torus knots

We compute the $E_2$ page in the Rasmussen spectral sequence from triply graded to $\mathfrak{gl}_N$ Khovanov--Rozansky stable homology of torus knots. This confirms a weak form of the conjecture of the second author, Oblomkov, and Rasmussen. The main tool is the link-splitting deformation, or $y$-ification, of link homology; in the $y$-ified context, the relevant Rasmussen spectral sequence collapses and we explicitly compute the $y$-ified $\mathfrak{gl}_N$ stable Khovanov--Rozansky homology of torus knots for all $N$.

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arXiv2024

Drinfeld centralizers and Rouquier complexes

The Drinfeld centralizer of a monoidal category $\mathcal{A}$ in a bimodule category $\mathcal{M}$ is the category $\mathcal{Z}(\mathcal{A},\mathcal{M})$ of objects in $\mathcal{M}$ for which the left and right actions by objects of $\mathcal{A}$ coincide, naturally. In this paper we study the interplay between Drinfeld centralizers of $\mathcal{A}$ and its homotopy category $\mathcal{K}^b(\mathcal{A})$, culminating with our ``lifting lemma,'' which provides a sufficient condition for an object of $\mathcal{Z}(\mathcal{A}, \mathcal{K}^b(\mathcal{M}))$ to lift to an object of $\mathcal{Z}(\mathcal{K}^b(\mathcal{A}), \mathcal{K}^b(\mathcal{M}))$. The central application of this lifting lemma is a proof of some folklore facts about conjugation by Rouquier complexes in the Hecke category: the centrality of the full twist, and related properties of half twists and Coxeter braids. We also prove stronger, homotopy coherent versions of these statements, stated using the notion of the $A_{\infty}$-Drinfeld centralizer, which we believe is new.

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arXiv2024

Envelopes and the bar complex

This paper is intended as a reference for some basic theory for dg categories and their bar complexes. Our modest goal is to carefully record the most important envelope operations can one perform on dg categories (in which one adjoins shifts, finite direct sums, or twists) and the inescapable sign rules that appear when combining these with opposite categories, tensor products, and the bar resolution. An appendix collects some theory of categorical idempotents that is useful when discussing bar complexes.

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arXiv2024

Khovanov-Rozansky homology of Coxeter knots and Schröder polynomials for paths under any line

We introduce a family of generalized Schröder polynomials $S_τ(q,t,a)$, indexed by triangular partitions $τ$ and prove that $S_τ(q,t,a)$ agrees with the Poincaré series of the triply graded Khovanov-Rozansky homology of the Coxeter knot $K_τ$ associated to $τ$. 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 Schröder 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 $(q,t)$-Schröder theorem.

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arXiv2024

Bordered invariants from Khovanov homology

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.

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arXiv2024

Tautological classes and symmetry in Khovanov-Rozansky homology

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.

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arXiv2023

On unification of colored annular sl(2) knot homology

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.

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arXiv2022

A Kirby color for Khovanov homology

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.

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arXiv2021

Link splitting deformation of colored Khovanov--Rozansky homology

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.

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arXiv2021

A skein relation for singular Soergel bimodules

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 $\mathfrak{sl}_2$.

math.QA↗

arXiv2021

The Fukaya category of the pillowcase, traceless character varieties, and Khovanov Cohomology

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.

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arXiv2020

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. 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.

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arXiv2020

Ext-enhanced monoidal Koszul duality for $\mathrm{GL}_2$

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.

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arXiv2020

Serre duality for Khovanov-Rozansky homology

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 Kálmán.

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arXiv2020

Constructing categorical idempotents

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.

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arXiv2020

Homological perturbation theory with curvature

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.

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arXiv2019

Torus link homology

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

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arXiv2019

On the functoriality of sl(2) tangle homology

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.

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