q-Rationals, link invariants and webs
We investigate the role of q-rationals in the context of link invariants. This leads us to introduce and study web categories that q-deform Deligne's categories.
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arXiv subjects
Hoel Queffelec
Publications and source records attributed to Hoel Queffelec.
Extriangulated structures from stability conditions and application to braid representations
We use the notion of Bridgeland stability condition and its associated metric to endow triangulated categories with extriangulated structures and study their extriangulated Grothendieck groups. This study is motivated by Khovanov-Seidel's categorification of the Burau representation, from which can be extracted the Lawrence-Krammer-Bigelow representation, providing a relation between these two faithful representations of the braid groups.
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A fast algorithm for the Hecke representation of the braid group, and applications to the computation of the HOMFLY-PT polynomial and the search for interesting braids
Knot theory is an active field of mathematics, in which combinatorial and computational methods play an important role. One side of computational knot theory, that has gained interest in recent years, both for complexity analysis and practical algorithms, is quantum topology and the computation of topological invariants issued from the theory. In this article, we leverage the rigidity brought by the representation-theoretic origins of the quantum invariants for algorithmic purposes. We do so by exploiting braids and the algebraic properties of the braid group to describe, analyze, and implement a fast algorithm to compute the Hecke representation of the braid group. We apply this construction to design a parameterized algorithm to compute the HOMFLY-PT polynomial of knots, and demonstrate its interest experimentally. Finally, we combine our fast Hecke representation algorithm with Garside theory, to implement a reservoir sampling search and find non-trivial braids with trivial Hecke representations with coefficients in $\mathbb{Z}/p\mathbb{Z}$. We find several such braids, in particular proving that the Hecke representation of $B_5$ with $\mathbb{Z}/2\mathbb{Z}$ coefficients is non-faithful, a previously unknown fact.
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Some remarks about the faithfulness of the Burau representation of Artin--Tits groups
We discuss the extension of the faithfulness question for the Burau representation of braid groups to the case of Artin--Tits groups. We prove that the Burau representation is not faithful in affine type $\tilde{A_3}$, and not faithful over several finite rings in type $D_4$, using an algorithmic approach based on categorical methods that generalize Bigelow's curve strategy outside of type $A$.
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Reidemeister's theorem using transversality
We recover Reidemeister's theorem using smooth functions and transversality.
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Khovanov-Seidel braid representation
These are lecture notes from a lecture series given at CIRM in the Fall 2023. They give a down-to-earth introduction to Khovanov and Seidel's categorical representation of Artin-Tits groups, emphasizing the fact that it is all explicitly computable. Several prospective applications to (geometric) group theory are mentioned.
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Gl2 Foam Functoriality and Skein Positivity
We prove full functoriality of Khovanov homology for tangled framed gl2 webs. We use this functoriality result to prove a strong positivity result for (orientable) surface skein algebras. The argument goes categorical and consists in proving that so-called linear complexes are stable under superposition.
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Movie moves for framed foams from multijet transversality
We use multijet transversality techniques to give a presentation by generators and relations of the categories of framed tangled webs and foams.
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Polynomial link invariants and quantum algebras
The definition of the Jones polynomial in the 80's gave rise to a large family of so-called quantum link invariants, based on quantum groups. These quantum invariants are all controlled by the same two-variable invariant (the HOMFLY-PT polynomial), which also specializes to the older Alexander polynomial. Building upon quantum Schur--Weyl duality and variants of this phenomenon, I will explain an algebraic setup that allows for global definitions of these quantum polynomials, and discuss extensions of these quantum objects designed to encompass all of the mentioned invariants, including the HOMFLY-PT polynomial.
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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 Thurston. 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.
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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 greater than or equal to 2, 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.
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Annular Evaluation and Link Homology
We use categorical annular evaluation to give a uniform construction of both $\mathfrak{sl}_n$ and HOMFLYPT Khovanov-Rozansky link homology, as well as annular versions of these theories. Variations on our construction yield $\mathfrak{gl}_{-n}$ link homology, i.e. a link homology theory associated to the Lie superalgebra $\mathfrak{gl}_{0|n}$, both for links in $S^3$ and in the thickened annulus. In the $n=2$ case, this produces a categorification of the Jones polynomial that we show is distinct from Khovanov homology, and gives a finite-dimensional categorification of the colored Jones polynomial. This behavior persists for general $n$. Our approach yields simple constructions of spectral sequences relating these theories, and emphasizes the roles of super vector spaces, categorical traces, and current algebras in link homology.
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An equivalence between truncations of categorified quantum groups and Heisenberg categories
We introduce a simple diagrammatic 2-category $\mathscr{A}$ that categorifies the image of the Fock space representation of the Heisenberg algebra and the basic representation of $\mathfrak{sl}_\infty$. We show that $\mathscr{A}$ is equivalent to a truncation of the Khovanov--Lauda categorified quantum group $\mathscr{U}$ of type $A_\infty$, and also to a truncation of Khovanov's Heisenberg 2-category $\mathscr{H}$. This equivalence is a categorification of the principal realization of the basic representation of $\mathfrak{sl}_\infty$. As a result of the categorical equivalences described above, certain actions of $\mathscr{H}$ induce actions of $\mathscr{U}$, and vice versa. In particular, we obtain an explicit action of $\mathscr{U}$ on representations of symmetric groups. We also explicitly compute the Grothendieck group of the truncation of $\mathscr{H}$. The 2-category $\mathscr{A}$ can be viewed as a graphical calculus describing the functors of $i$-induction and $i$-restriction for symmetric groups, together with the natural transformations between their compositions. The resulting computational tool is used to give simple diagrammatic proofs of (apparently new) representation theoretic identities.
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Braid groups of type ADE, Garside monoids, and the categorified root lattice
We study Artin-Tits braid groups $\mathbb{B}_W$ of type ADE via the action of $\mathbb{B}_W$ on the homotopy category $\mathcal{K}$ of graded projective zigzag modules (which categorifies the action of the Weyl group $W$ on the root lattice). Following Brav-Thomas, we define a metric on $\mathbb{B}_W$ induced by the canonical $t$-structure on $\mathcal{K}$, and prove that this metric on $\mathbb{B}_W$ agrees with the word-length metric in the canonical generators of the standard positive monoid $\mathbb{B}_W^+$ of the braid group. We also define, for each choice of a Coxeter element $c$ in $W$, a baric structure on $\mathcal{K}$. We use these baric structures to define metrics on the braid group, and we identify these metrics with the word-length metrics in the Birman-Ko-Lee/Bessis dual generators of the associated dual positive monoid $\mathbb{B}_{W.c}^\vee$. As consequences, we give new proofs that the standard and dual positive monoids inject into the group, give linear-algebraic solutions to the membership problem in the standard and dual positive monoids, and provide new proofs of the faithfulness of the action of $\mathbb{B}_W$ on $\mathcal{K}$. Finally, we use the compatibility of the baric and $t$-structures on $\mathcal{K}$ to prove a conjecture of Digne and Gobet regarding the canonical word-length of the dual simple generators of ADE braid groups.
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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.
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Mixed quantum skew Howe duality and link invariants of type A
We define a ribbon category $\mathsf{Sp}(β)$, depending on a parameter $β$, which encompasses Cautis, Kamnitzer and Morrison's spider category, and describes for $β=m-n$ the monoidal category of representations of $U_q(\mathfrak{gl}_{m|n})$ generated by exterior powers of the vector representation and their duals. We identify this category $\mathsf{Sp}(β)$ with a direct limit of quotients of a dual idempotented quantum group $\dot{\mathsf{U}}_q(\mathfrak{gl}_{r+s})$, proving a mixed version of skew Howe duality in which exterior powers and their duals appear at the same time. We show that the category $\mathsf{Sp}(β)$ gives a unified natural setting for defining the colored $\mathfrak{gl}_{m|n}$ link invariant (for $β=m-n$) and the colored HOMFLY-PT polynomial (for $β$ generic).
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Sutured annular Khovanov-Rozansky homology
We introduce an sl(n) homology theory for knots and links in the thickened annulus. To do so, we first give a fresh perspective on sutured annular Khovanov homology, showing that its definition follows naturally from trace decategorifications of enhanced sl(2) foams and categorified quantum gl(m), via classical skew Howe duality. This framework then extends to give our annular sl(n) link homology theory, which we call sutured annular Khovanov-Rozansky homology. We show that the sl(n) sutured annular Khovanov-Rozansky homology of an annular link carries an action of the Lie algebra sl(n), which in the n=2 case recovers a result of Grigsby-Licata-Wehrli.
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A note on the gl(m|n) link invariants and the HOMFLY-PT polynomial
We present a short and unified representation-theoretical treatment of type A link invariants (that is, the HOMFLY-PT polynomials, the Jones polynomial, the Alexander polynomial and, more generally, the gl(m|n) quantum invariants) as link invariants with values in the quantized oriented Brauer category.
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