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- The universal sl(3)-link homologyPublication . Mackaay, Marco; Vaz, PedroWe define the universal sl(3)-link homology, which depends on 3 parameters, following Khovanov's approach with foams. We show that this 3-parameter link homology, when taken with complex coefficients, can be divided into 3 isomorphism classes. The first class is the one to which Khovanov's original sl(3)-link homology belongs, the second is the one studied by Gornik in the context of matrix factorizations and the last one is new. Following an approach similar to Gornik's we show that this new link homology can be described in terms of Khovanov's original sl(2)-link homology.
- The foam and the matrix factorization sl(3) link homologies are equivalentPublication . Mackaay, Marco; Vaz, PedroWe prove that the universal rational sl(3) link homologies which were constructed by Khovanov in [3] and the authors in [7], using foams, and by Khovanov and Rozansky in [4], using matrix factorizations, are naturally isomorphic as projective functors from the category of links and link cobordisms to the category of bigraded vector spaces.
- A remark on Rasmussen's invariant of knotsPublication . Mackaay, Marco; Turner, Paul; Vaz, PedroWe show that Rasmussen's invariant of knots, which is derived from Lee's variant of Khovanov homology, is equal to an analogous invariant derived from certain other filtered link homologies.
- The foam and the matrix factorization sl3 link homologies are equivalentPublication . Mackaay, Marco; Vaz, PedroWe prove that the foam and matrix factorization universal rational sl3 link homologies are naturally isomorphic as projective functors from the category of link and link cobordisms to the category of bigraded vector spaces.
- The 1,2-coloured HOMFLY-PT link homologyPublication . Mackaay, Marco; Stosic, Marko; Vaz, PedroIn this paper we define the 1,2-coloured HOMFLY-PT triply graded link homology and prove that it is a link invariant. We also conjecture on how to generalize our construction for arbitrary colours.
- Evaluation birepresentations of affine type a soergel bimodulesPublication . Mackaay, Marco; Miemietz, Vanessa; Vaz, PedroIn this paper, we use Soergel calculus to define a monoidal functor, called the evaluation functor, from extended affine type A Soergel bimodules to the homotopy category of bounded complexes in finite type A Soergel bimodules. This functor categorifies the well-known evaluation homomorphism from the extended affine type A Hecke algebra to the finite type A Hecke algebra. Through it, one can pull back the triangulated birepresentation induced by any finitary birepresentation of finite type A Soergel bimodules to obtain a triangulated birepresentation of extended affine type A Soergel bimodules. We show that if the initial finitary birepresentation in finite type A is a cell birepresentation, the evaluation birepresentation in extended affine type A has a finitary cover, which we illustrate by working out the case of cell birepresentations with subregular apex in detail. (c) 2023 Elsevier Inc. All rights reserved.
- sl(N)-link homology (N >= 4) using foams and the Kapustin-Li formulaPublication . Mackaay, Marco; Stosic, Marko; Vaz, PedroWe use foams to give a topological construction of a rational link homology categorifying the sl(N) link invariant, for N >= 4. To evaluate closed foams we use the Kapustin-Li formula adapted to foams by Khovanov and Rozansky [9]. We show that for any link our homology is isomorphic to the Khovanov-Rozansky [11] homology.