Supercategorification and Khovanov-like tangle invariants

Details

Supervisors

Dos Santos Santana Forte Vaz, Pedro

Faculty

Faculté des sciences

Degree label

Master [120] en sciences mathématiques

Abstract

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.