Fix an integer N ≥ 2. To each diagram of a link colored by 1, . . . , N , we associate a chain complex of graded matrix factorizations. We prove that the homotopy type of this chain complex is invariant under Reidemeister moves. When every component of the link is colored by 1, this chain complex is isomorphic to the chain complex defined by Khovanov and Rozansky in [19]. The homology of this chain complex decategorifies to the Reshetikhin-Turaev sl(N ) polynomial of links colored by exterior powers of the defining representation.
Abstract. Fix an integer N ≥ 2. To each diagram of a link colored by 1, . . . , N , we associate a chain complex of graded matrix factorizations. We prove that the homotopy type of this chain complex is invariant under Reidemeister moves. When every component of the link is colored by 1, this chain complex is isomorphic to the chain complex defined by Khovanov and Rozansky in [19]. The homology of this chain complex decategorifies to the Reshetikhin-Turaev sl(N ) polynomial of links colored by exterior powers of the defining representation.
We prove the quantum filtration on the Khovanov-Rozansky link cohomology Hp with a general degree (n + 1) monic potential polynomial p(x) is invariant under Reidemeister moves, and construct a spectral sequence converging to Hp that is invariant under Reidemeister moves, whose E 1 term is isomorphic to the Khovanov-Rozansky sl(n)-cohomology Hn. Then we define a generalization of the Rasmussen invariant, and study some of its properties. We also discuss relations between upper bounds of the self-linking number of transversal links in standard contact S 3 .2000 Mathematics Subject Classification. Primary 57M25, Secondary 57R17. Key words and phrases. knot homology, matrix factorization, transversal knot.Proof. Straightforward. Definition 2.8. Let (M, F ) be a filtered R-module. F is called faithful if deg (r · m) = deg r + deg m, for any non-zero elements r ∈ R, m ∈ M . In this case, M is said to be a faithfully filtered R-module.Lemma 2.9. If M and N are both faithfully filtered R-modules, then so is M ⊕ N .We usually write such a matrix factorization M asFollowing [11], we denote by M 1 the filtered matrix factorizationby M * the filtered matrix factorization
In this sequel to [A colored sl(N )-homology for links in S 3 , preprint (2009), arXiv:0907.0695v5], we construct an equivariant colored sl(N )-homology for links, which generalizes both the above mentioned paper and the paper by [D. Krasner, Equivariant sl(n)-link homology, preprint (2008), arXiv:0804.3751v2]. The construction is a straightforward generalization of the paper, [A colored sl(N )-homology for links in S 3 ]. The proof of invariance is based on a simple observation which allows us to translate the proof in the above mentioned paper into the new setting.As an application, we prove that deformations over C of the colored sl(N )-homology are link invariants. We also construct a spectral sequence connecting the colored sl(N )-homology to its deformations over C, which generalizes the spectral sequence given in the papers by [B. Gornik, Note on Khovanov link cohomology, preprint (2004), arXiv:math.QA/0402266 and E. Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197(2) (2005) 554-586].
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