The universal sl3–link homology

By Marco Mackaay, Pedro Vaz. 2006

71 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/014917ef46cc73d41e90ecbc3de0f49ce1d1ffd8

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Man and machine thinking about the smooth 4-dimensional Poincaré conjecture (2009, background): “Khovanov homology (or its variants categorifying the sl3 polynomial [27], [33], [35] or sln polynomials [29], [30], [43]) may hold promise for establishing that a link L is not slice.”
sl.N/-link homology (N 4) using foams and the Kapustin-Li formula (2007, methodology): “Mackaay and Vaz [12] generalized Khovanov’s results to obtain the universal sl.”
Some differentials on Khovanov–Rozansky homology (2006, background): “ed by the multiplicities of the roots of p′(x). (This idea has its source in the work of Gornik [4]. Although we will not pursue it here, some supporting evidence has been provided by Mackaay and Vaz [16].) In particular, it seems unlikely that the set of all homologies Hp(L,i), p∈ Q[x] contains mor”
Braid cobordisms, triangulated categories, and flag varieties (2006, methodology): “Following [ MV ], one should also obtain a categorification of the braid group action on Inv(V ⊗n ⊗ W), by adding dummy strands.”
A combinatorial approach to functorial quantum slk knot invariants (2007, background): “ation theoretical background for the very recent generalisation [19] of [16] to arbitrary k. Roughly speaking, a foam is a morphism between certain trivalent graphs (for a precise deﬁnition see [16], [20], [19]). Khovanov associated to each special trivalent graph a graded vector space and to any fo”
Deformations of colored sl(N) link homologies via foams (2015, background): “f a link decomposes into the direct sum of sl1 homologies of various sub-links, which are always 1-dimensional. Other deformations have been studied for N= 2 by Khovanov [Kh2] and N= 3 by Mackaay-Vaz [MV]. In this paper, we prove a vast generalization of these results, showing that the deformation o”
The $\mathfrak{sl}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality (2014, background): “Khovanov’s sl3 link invariant [23] admits a similar description [42], [45].”
Functoriality of colored link homologies (2017, methodology): “ant, or universal, link homology can be traced back to Bar-Natan [BN05] and Khovanov [Kho06] and it encapsulated both Khovanov homology and Lee’s deformation. The case N= 3 was treated by Mackaay{Vaz [MV07]. The extension to higher rank is due to Krasner [Kra10] in the uncolored case, and due to Wu ”
Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m) (2012, methodology): “These relations are similar to the ones of [53] in the sl3 case, but in the sl2 case we obtain relations that slightly extend both Blanchet’s [4] and Clark-Morrison-Walker [21] models.”