Publications
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Preprint (2025), submitted
A local $\mathfrak{gl}_{1|1}$-action on odd Khovanov homology
arXiv
We show that odd Khovanov homology carries an action of the super Lie algebra $\mathfrak{gl}_{1|1}$, given extra choice of markings on the link. Moreover, we show that this action arises from an action on super $\mathfrak{gl}_{2}$-foams, in the extended-TQFT framework developed by the second author and Vaz; in particular, it extends to tangles. Finally, we relate the action to torsion $\mathbb{Z}/n\mathbb{Z}$ in pretzel links $P(n,n,-n)$. In particular, this shows that all torsion can appear in odd Khovanov homology.
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Preprint (2025), submitted
A basis and Schur–Weyl duality for the loop Hecke algebra
arXiv
The loop Hecke algebra is a generalization of the Hecke algebra to the loop braid group, introduced by Damiani, Martin and Rowell. We give a new presentation of the loop Hecke algebra provided a mild condition on the parameter and give a basis. We use higher linear rewriting theory to show linear independence and the combinatorics of Dyck paths to compute the cardinality of the basis. This yields a conjecture of Damiani-Martin-Rowel. We also give a representation theoretic interpretation of the loop Hecke algebra in terms of (non-semisimple) Schur-Weyl duality involving the negative half of quantum $\mathfrak{gl}_{1|1}$.
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Advances in Mathematics (2026), preprint (2025)
Rewriting modulo in diagrammatic algebras and application to categorification
arXiv
journal
We develop a rewriting theory modulo suitable for higher linear structures. In particular, the theory is suited for diagrammatic algebras as they appear in categorification, representation theory and quantum topology. As an application, we use the theory to prove the basis conjecture of a certain super-2-category related to odd Khovanov homology. Our approach combines linear rewriting, higher rewriting and rewriting modulo. For diagrammatic algebras, the modulo rules typically capture a categorical property, such as pivotality. In the process, we revisit the foundations of these theories, including the notion of confluence. Other important tools include termination rules that depend on contexts, rewriting modulo invertible scalars, and a method to classify branchings modulo. This article includes an introduction to rewriting theory for non-experts.
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Quantum Topology (2026), preprint (2023)
Odd Khovanov homology and higher representation theory
arXiv
journal
We introduce a super analogue of $\mathfrak{gl}_2$-foams, and use it to define an invariant of oriented tangles, shown to coincide with odd Khovanov homology when restricted to links. We then define a supercategorification of the $q$-Schur algebra of level 2 and realize our construction as a certain super-2-representation. This gives a representation-theoretic construction of odd Khovanov homology, where signs naturally arise as a byproduct of the super-2-categorical structure. In the process, we define a tensor product for chain complexes in super-2-categories, suitably compatible with homotopies. This could be of independent interest.
Thesis
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PhD thesis (2024)
Odd Khovanov homology, higher representation theory and higher rewriting theory
arXiv
university repository
This thesis is devoted to the fields of quantum topology and rewriting theory, and their surprising interconnections. In the first part of the thesis, we develop a higher representation theoretic approach to odd Khovanov homology; this is the content of arXiv:2311.14394. One of the essential ingredients is a certain graded-2-category of graded $\mathfrak{gl}_2$-foams. In the second part of the thesis, we develop a rewriting theory suitable for higher algebras and their super or graded analogues, and use it to show a basis theorem for graded $\mathfrak{gl}_2$-foams. These techniques have the potential to be applied to a wide variety of contexts. Both parts of the thesis can be read independently. Each has its own comprehensive introduction, allowing experts from one field to get acquainted with the other field.
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Master thesis (2020)
Supercategorification and Khovanov-like tangle invariants
university repository
The Jones polynomial is an invariant of links arising as a quantum invariant through the representation theory of the quantum group Uq(sl2). In 2000, Khovanov showed that it can also be obtained as the graded Euler characteristic of a certain link homology, giving a strictly stronger invariant of links. It also detects higher dimensional information, as it defines a functor from the category of links in 3-space and cobordisms to the category of chain complexes up to homotopy and chain maps. In 2008, Webster gave a representation theory construction of Khovanov homology by categorifying Uq(sl2) and its representations. Lauda, Queffelec and Rose later showed an alternative construction using a categorification of Uq(slm). In 2013 Oszváth, Rasmussen and Szabó constructed another categorification of the Jones polynomial called odd Khovanov homology using exterior algebras, distinct from the usual Khovanov homology. In march 2020, Naisse and Putyra extended this construction to tangles. At the time being though, there is no representation theory construction for odd Khovanov homology, and its functoriality is yet to be shown. In 2019, Vaz used superalgebras to give a Khovanov type invariant of tangles, conjectured to coincide with odd Khovanov homology when restricted to links. In this thesis, we continue this work and study the problem of giving a construction of odd Khovanov homology using higher representation theory. More precisely, we categorify a Schur algebra of level two using a 2-supercategory. Constructing a tangle invariant via this 2-supercategory calls for the appropriate notion of tensor product of chain complexes, different from the usual since the maps we use to construct the differentials are not of parity zero as in the usual constructions of categories of complexes over supercategories. Therefore, we define a new extension of the Koszul rule and show that the tensor product of complexes leaves homotopy types invariant. Using this and the properties of the 2-supercategory, we associate a chain complex to any tangle, and prove that its homotopy type is a link invariant. We conjecture that it coincides with odd Khovanov homology when restricted to links. We hope that our construction will give the tools to prove the functoriality of odd Khovanov homology.