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Anna Beliakova

Publications and source records attributed to Anna Beliakova.

At least 19 recordsLinked to original sources
arXiv2025

Algebraic Presentation of $4$-Dimensional $2$-Handlebodies and $3$-Dimensional Cobordisms

In this paper, we give a new direct proof of a result by Bobtcheva and Piergallini that provides finite algebraic presentations of two categories, denoted $3\mathrm{Cob}$ and $4\mathrm{HB}$, whose morphisms are manifolds of dimension $3$ and $4$, respectively. More precisely, $3\mathrm{Cob}$ is the category of connected oriented $3$-dimensional cobordisms between connected surfaces with connected boundary, while $4\mathrm{HB}$ is the category of connected oriented $4$-dimensional $2$-handlebodies up to $2$-deformations. For this purpose, we explicitly construct the inverse of the functor $Φ: 4\mathrm{Alg} \to 4\mathrm{HB}$, where $4\mathrm{Alg}$ denotes the free monoidal category generated by a Bobtcheva--Piergallini Hopf algebra. As an application, we deduce an algebraic presentation of $3\mathrm{Cob}$ and show that it is equivalent to the one conjectured by Habiro.

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arXiv2025

Quantum Invariants of Ribbon Surfaces in $4$-Dimensional $2$-Handlebodies

We use unimodular ribbon categories to construct quantum invariants of ribbon surfaces in $4$-dimensional $2$-handlebodies up to $1$-isotopy. In the process, we recover invariants due to Bobtcheva-Messia, Broda-Petit, Gainutdinov-Geer-Patureau-Runkel (in collaboration with the second author), and Lee-Yetter. Our approach does not assume semisimplicity, and is based on a generalization of the Reshetikhin-Turaev functor to the category of labeled Kirby graphs which also yields invariants of framed links in the boundary of $4$-dimensional $2$-handlebodies up to $2$-deformations. The setup is very flexible, and allows for several different constructions, using central elements satisfying equations introduced by Hennings and Bobtcheva-Messia, modified traces, and modules over Frobenius algebras satisfying conditions dictated by the diagrammatic calculus for embedded surfaces developed by Hughes, Kim, and Miller.

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arXiv2025

A proof of Dunfield-Gukov-Rasmussen Conjecture

In 2005 Dunfield, Gukov and Rasmussen conjectured an existence of the spectral sequence from the reduced triply graded Khovanov-Rozansky homology of a knot to its knot Floer homology defined by Ozsváth and Szabó. The main result of this paper is a proof of this conjecture. For this purpose, we construct a bigraded spectral sequence from the $\mathfrak{gl}_0$ homology constructed by the last two authors to the knot Floer homology. Using the fact that the $\mathfrak{gl}_0$ homology comes equipped with a spectral sequence from the reduced triply graded homology, we obtain our main result. The first spectral sequence is of Bockstein type and comes from a subtle manipulation of coefficients. The main tools are quantum traces of foams and of singular Soergel bimodules and a $\mathbb Z$-valued cube of resolutions model for knot Floer homology originally constructed by Ozsváth and Szabó over the field of two elements. As an application, we deduce that the $\mathfrak{gl}_0$ homology as well as the reduced triply graded Khovanov-Rozansky one detect the unknot, the two trefoils, the figure eight knot and the cinquefoil.

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arXiv2024

Abelian TQFTS and Schrödinger local systems

We construct an action of 3-cobordisms on the finite dimensional Schrödinger representations of the Heisenberg group by Lagrangian correspondences. In addition, we review the construction of the abelian Topological Quantum Field Theory (TQFT) associated with a $q$-deformation of $U(1)$ for any root of unity $q$. We prove that for3-cobor\-disms compatible with Lagrangian correspondences, there is a normalization of the associated Schrödinger bimodule action that reproduces the abelian TQFT. The full abelian TQFT provides a projective representation of the mapping class group $\mathrm{Mod}(Σ)$ on the Schrödinger representation,which is linearizable at odd root of 1. Motivated by homology of surface configurations with Schrödinger representation as local coefficients, we define another projective action of $\mathrm{Mod}(Σ)$ on Schrödinger representations. We show that the latter is not linearizable by identifying the associated 2-cocycle.

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arXiv2023

Refined Bobtcheva-Messia Invariants of 4-Dimensional 2-Handlebodies

In this paper we refine our recently constructed invariants of $4$-dimensional $2$-handlebodies up to $2$-deformations. More precisely, we define invariants of pairs of the form $(W,ω)$, where $W$ is a $4$-dimensional $2$-handlebody, $ω$ is a relative cohomology class in $H^2(W,\partial W;G)$, and $G$ is an abelian group. The algebraic input required for this construction is a unimodular ribbon Hopf $G$-coalgebra. We study these refined invariants for the restricted quantum group $U = U_q \mathfrak{sl}_2$ at a root of unity $q$ of even order, and for its braided extension $\tilde{U} = \tilde{U}_q \mathfrak{sl}_2$, which fits in this framework for $G=\mathbb{Z}/2\mathbb{Z}$, and we relate them to our original invariant. We deduce decomposition formulas for the original invariants in terms of the refined ones, generalizing splittings of the Witten-Reshetikhin-Turaev invariants with respect to spin structures and cohomology classes. Moreover, we identify our non-refined invariant associated with the small quantum group $\bar{U} = \bar{U}_q \mathfrak{sl}_2$ at a root of unity $q$ whose order is divisible by 4 with the refined one associated with the restricted quantum group $U$ for the trivial cohomology class $ω=0$.

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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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arXiv2023

Kerler-Lyubashenko Functors on 4-Dimensional 2-Handlebodies

We construct a braided monoidal functor $J_4$ from Bobtcheva and Piergallini's category $4\mathrm{HB}$ of connected 4-dimensional 2-handlebodies (up to 2-deformations) to an arbitrary unimodular ribbon category $\mathcal{C}$, which is not required to be semisimple. The main example of target category is provided by $H$-mod, the category of left modules over a unimodular ribbon Hopf algebra $H$. The source category $4\mathrm{HB}$ is freely generated, as a braided monoidal category, by a BPH algebra (short for Bobtcheva-Piergallini Hopf algebra), and this is sent by the Kerler-Lyubashenko functor $J_4$ to the end $\int_{X \in \mathcal{C}} X \otimes X^*$ in $\mathcal{C}$, which is given by the adjoint representation in the case of $H$-mod. When $\mathcal{C}$ is factorizable, we show that the construction only depends on the boundary and signature of handlebodies, and thus projects to a functor $J_3^σ$ defined on Kerler's category $3\mathrm{Cob}^σ$ of connected framed 3-dimensional cobordisms. When $H^*$ is not semisimple and $H$ is not factorizable, our functor $J_4$ has the potential of detecting diffeomorphisms that are not 2-deformations.

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arXiv2023

Current algebras and categorified quantum groups

We identify the trace, or 0th Hochschild homology, of type ADE categorified quantum groups with the corresponding current algebra of the same type. To prove this, we show that 2-representations defined using categories of modules over cyclotomic (or deformed cyclotomic) quotients of KLR-algebras correspond to local (or global) Weyl modules. We also investigate the implications for centers of categories in 2-representations of categorified quantum groups.

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arXiv2021

Cyclotomic expansions for $\mathfrak{gl}_N$ knot invariants via interpolation Macdonald polynomials

In this paper we construct a new basis for the cyclotomic completion of the center of the quantum $\mathfrak{gl}_N$ in terms of the interpolation Macdonald polynomials. Then we use a result of Okounkov to provide a dual basis with respect to the quantum Killing form (or Hopf pairing). The main applications are: 1) cyclotomic expansions for the $\mathfrak{gl}_N$ Reshetikhin--Turaev link invariants and the universal $\mathfrak{gl}_N$ knot invariant; 2) an explicit construction of the unified $\mathfrak{gl}_N$ invariants for integral homology 3-spheres using universal Kirby colors. These results generalize those of Habiro for $\mathfrak{sl}_2$. In addition, we give a simple proof of the fact that the universal $\mathfrak{gl}_N$ invariant of any evenly framed link and the universal $\mathfrak{sl}_N$ invariant of any $0$-framed algebraically split link are $Γ$-invariant, where $Γ=Y/2Y$ with the root lattice $Y$.

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arXiv2020

Modified trace is a symmetrised integral

A modified trace for a finite k-linear pivotal category is a family of linear forms on endomorphism spaces of projective objects which has cyclicity and so-called partial trace properties. We show that a non-degenerate modified trace defines a compatible with duality Calabi-Yau structure on the subcategory of projective objects. The modified trace provides a meaningful generalisation of the categorical trace to non-semisimple categories and allows to construct interesting topological invariants. We prove, that for any finite-dimensional unimodular pivotal Hopf algebra over a field k, a modified trace is determined by a symmetric linear form on the Hopf algebra constructed from an integral. More precisely, we prove that shifting with the pivotal element defines an isomorphism between the space of right integrals, which is known to be 1-dimensional, and the space of modified traces. This result allows us to compute modified traces for all simply laced restricted quantum groups at roots of unity.

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arXiv2020

Non-semisimple invariants and Habiro's series

In this paper we establish an explicit relationship between Habiro's cyclotomic expansion of the colored Jones polynomial (evaluated at a p-th root of unity) and the Akutsu-Deguchi-Ohtsuki (ADO) invariants of the double twist knots. This allows us to compare the Witten-Reshetikhin-Turaev (WRT) and Costantino-Geer-Patureau (CGP) invariants of 3-manifolds obtained by 0-surgery on these knots. The difference between them is determined by the p-1 coefficient of the Habiro series. We expect these to hold for all Seifert genus 1 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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arXiv2018

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 $\mathbf{C}$ and endobifunctor $Σ\colon \mathbf C \to\mathbf C$. For a graded linear bicategory and a fixed invertible parameter $q$, we quantize this theory by using the endofunctor $Σ_q$ such that $Σ_q α:=q^{-°α}Σα$ 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 $\mathcal U_q(\mathfrak{sl}_2)$, 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 $q$.

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arXiv2017

Logarithmic Hennings invariants for restricted quantum sl(2)

We construct a Hennings type logarithmic invariant for restricted quantum $\mathfrak{sl}(2)$ at a $2\mathsf{p}$-th root of unity. This quantum group $U$ is not braided, but factorizable. The invariant is defined for a pair: a 3-manifold $M$ and a colored link $L$ inside $M$. The link $L$ is split into two parts colored by central elements and by trace classes, or elements in the $0^{\text{th}}$ Hochschild homology of $U$, respectively. The two main ingredients of our construction are the universal invariant of a string link with values in tensor powers of $U$, and the modified trace introduced by the third author with his collaborators and computed on tensor powers of the regular representation. Our invariant is a colored extension of the logarithmic invariant constructed by Jun Murakami.

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arXiv2015

Cyclicity for categorified quantum groups

We equip the categorified quantum group attached to a KLR algebra and an arbitrary choice of scalars with duality functor which is cyclic, that is, such that f=f^** for all 2-morphisms f. This is accomplished via a modified diagrammatic formalism.

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arXiv2014

Spin Modular Categories

Modular categories are a well-known source of quantum 3-manifold invariants. In this paper we study structures on modular categories which allow to define refinements of quantum 3-manifold invariants involving cohomology classes or generalized spin and complex spin structures. A crucial role in our construction is played by objects which are invertible under tensor product. All known examples of cohomological or spin type refinements of the Witten-Reshetikhin-Turaev 3-manifold invariants are special cases of our construction. In addition, we establish a splitting formula for the refined invariants, generalizing the well-known product decomposition of quantum invariants into projective ones and those determined by the linking matrix.

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arXiv2014

Trace as an alternative decategorification functor

Categorification is a process of lifting structures to a higher categorical level. The original structure can then be recovered by means of the so-called "decategorification" functor. Algebras are typically categorified to additive categories with additional structure and decategorification is usually given by the (split) Grothendieck group. In this expository article we study an alternative decategorification functor given by the trace or the zeroth Hochschild--Mitchell homology. We show that this form of decategorification endows any 2-representation of the categorified quantum sl(n) with an action of the current algebra U(sl(n)[t]) on its center.

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