Marco Mackaay

The universal sl3–link homology

We define the universal sl3 –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 sl3 –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 sl2 –link homology.

sl.N/-link homology (N 4) using foams and the Kapustin-Li formula

We use foams to give a topological construction of a rational link homology categorifying the slN link invariant, for N>3. To evaluate closed foams we use the Kapustin-Li formula adapted to foams by Khovanov and Rozansky. We show that for any link our homology is isomorphic to Khovanov and Rozanskys.

A remark on rasmussen's invariant of knots

We show that Rasmussens invariant of knots, which is derived from Lees variant of Khovanov homology, is equal to an analogous invariant derived from certain other filtered link homologies.

The 1,2-coloured HOMFLY-PT link homology

In 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.

THE sl3-WEB ALGEBRA

In this paper we use Kuperbergs sl3-webs and Khovanovs sl3-foams to define a new algebra K S , which we call the sl3-web algebra. It is the sl3 analogue of Khovanovs arc algebra. We prove that K S is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of q-skew Howe duality, which allows us to prove that K S is Morita equivalent to a cer- tain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group K ⊕ 0 (W S )Q(q), to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that K S is a graded cellular algebra.

Extended graphical calculus for categorified quantum sl(2)

We study the properties of the extended graphical calculus for categorified quantum

The foam and the matrix factorization sl3 link homologies are equivalent

sl(n)

The \(\mathfrak {sl}_{3}\)-web algebra

. The main results include proofs of Reidemeister 2 and Reidemeister 3-like moves involving strands corresponding to arbitrary thicknesses and arbitrary colors -- the results that were anounced in [M. Stosic: Indecomposable objects and Lusztigs canonical basis, Math. Res. Lett. 22, no. 1 (2015), 245-278].

The Diagrammatic Soergel Category and sl(N)-Foams, for N≥4

We 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.

Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification

In this paper we use Kuperbergs sl3-webs and Khovanovs sl3-foams to define a new algebra K S , which we call the sl3-web algebra. It is the sl3 analogue of Khovanovs arc algebra. We prove that K S is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of q-skew Howe duality, which allows us to prove that K S is Morita equivalent to a cer- tain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group K ⊕ 0 (W S )Q(q), to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that K S is a graded cellular algebra.