Note on Khovanov link cohomology

By Bojan Gornik. 2004

72 citations recorded by the source corpus.

We extend Lee's result on sl(2) Khovanov cohomology of a link L to the general sl(n) case: a filtered chain complex C(L) whose spectral sequence E_2 term equals Khovanov cohomology is exhibited. We also compute C(L)'s cohomology: it depends only on linking numbers of certain sublinks of L.

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

View Paper: https://www.semanticscholar.org/paper/495703c3ee52f8544840f9691d656fd8522697df

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

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Citations in the corpus

Deformations of colored sl(N) link homologies via foams (2015, methodology): “Lobb [Lo1, Lo2] has used Gornik’s generic deformation of slN link homology to define concordance invariants that are analogous to Rasmussen’s invariant.”
The universal sl3–link homology (2006, background): “ot take into account the internal grading of the Khovanov homology. We have tried to make the paper reasonably self-contained, but we do assume familiarity with the papers by Bar-Natan [1, 2], Gornik [4] and Khovanov [6, 7, 8]. 2. THE UNIVERSAL sl3-LINK HOMOLOGY Let Lbe a link in S3 and Da diagram o”
Functoriality of colored link homologies (2017, methodology): “provided by Bar-Natan{Morrison [BNM06]. The Khovanov{Rozansky link homologies of higher rank are subject to an even richer deformation theory, the exploration of which started with the work of Gornik [Gor04] and Mackaay{Vaz [MV07] in the uncolored case, and Wu [Wu12] in the colored case. These defor”
Holomorphic disks, link invariants and the multi-variable Alexander polynomial (2008, background): “raded object associated to a ﬁltered chain complex whose total homology is Z; and indeed, the ﬁltered chain homotopy type of this complex is a knot invariant, cf. [19], [24] (see also [12], [25], and [5] for corresponding results in Khovanov’s homology [8], [9]). Thus, there is a spectral sequence s”
A slice genus lower bound from sl(n) Khovanov–Rozansky homology (2007, methodology): “HKhi,jn(D) is an invariant of the knot and not just the diagram D. This homology theory is also functorial for knot cobordisms. Gornik has carried out for HKhn a direct analogue of Lee’s work on HKh2 [2], describing a perturbation with ﬁltered homology HKh′ n, which can be interpreted as the E∞ page”
The Superpolynomial for Knot Homologies (2005, methodology): “We are grateful to P. Etingof, B. Gornik, V. Kac, M. Khovanov, C. Manolescu, P. Ozsváth, A. Schwarz, C. Taubes, C. Vafa, and Z. Szabó for valuable discussions, and also to the referee for useful comments.”
Khovanov-Rozansky homology of two-bridge knots and links (2005, methodology): “Then the E1 term of Gornik’s spectral sequence would contain the generator a1 ⊗ X N−1, which must be killed by some generator with homological grading −3, and the q grading of this element would be at least 2N greater than the q-grading of a1 ⊗ X N−1.”
Homological algebra of knots and BPS states (2011, background): “ntials. Namely, in doubly-graded knot homologies constructed from matrix factorizations diﬀerentials that relate diﬀerent theories are in one-to-one correspondence with deformations of the potentials [51, 52, 53, 54, 55, 6]: (3.12) diﬀerentials on Hg,R ⇐⇒ deformations of W g,R For example, deformati”
Some differentials on Khovanov–Rozansky homology (2006, background): “The fact that the definition is interesting for other values of p was observed by Gornik [ 4 ].”