Matrix factorizations and link homology

By Mikhail Khovanov, L. Rozansky. 2004

617 citations recorded by the source corpus.

Author(s): Khovanov, Mikhail; Rozansky, Lev | Abstract: For each positive integer n the HOMFLY polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum sl(n). For each such n we build a doubly-graded homology theory of links with this polynomial as the Euler characteristic. The core of our construction utilizes the theory of matrix factorizations, which provide a linear algebra description of maximal Cohen-Macaulay modules on isolated hypersurface singularities.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9c5254fe0cf71102eb2ff50600102705678ceac8

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Citations in the corpus

Homological algebra of knots and BPS states (2011, methodology): “Conventions for the superpolynomial in the vector representation: The case r = 1 corresponds to the (ordinary) HOMFLY polynomial and the Khovanov-Rozansky homology.”
Categorifications from planar diagrammatics (2010, positive): “introduced in [ 29 ], and can be viewed as a categorification of the Ocneanu trace, thus a categorification of the HOMFLYPT polynomial.”
Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations (2010, background): “Matrix factorizations also appear prominently in the work of Khovanov and Rozansky [17] on link homology.”
On the computation of torus link homology (2016, methodology): “.00407v1 [math.GT] 1 Mar 2016 2 BEN ELIAS AND MATTHEW HOGANCAMP 1. INTRODUCTION Triply-graded Khovanov-Rozansky homology is a link homology theory which was originally introduced by Khovanov-Rozansky [KR08] using matrix factorizations, but was soon after reinterpreted by Khovanov [Kho07a] using the ”
Fivebranes and Knots (2011, methodology): “In one construction [21,22], the effective theory on Σ seems to be a Landau-Ginzburg B-model (so that the branes are matrix factorizations); in a second construction [23], the effective theory is a B-model with target space a certain Kahler manifold; other approaches [24, 25] are based on A-models.”
The $\mathfrak{sl}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality (2014, methodology): “Acknowledgements: We’d like to thank Christian Blanchet, Sabin Cautis, Ben Cooper, Matt Hogancamp, Mikhail Khovanov, Tony Licata, Scott Morrison, Lev Rozansky, and Hao Wu for helpful discussions, and Daniel Tubbenhauer for his careful reading of a preliminary draft of this paper.”
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.”
Fivebranes and 3-manifold homology (2016, methodology): “This can be illustrated with the following diagram: 6d (2, 0) theory on Σ× S1 ×M3 ↙ ↘ 3d N = 2 theory T [M3] twisted 3d N = 4 theory on M3 on S1 × Σ • UV: gauge theory • IR: Rozansky-Witten In the next sections we will consider these two different points of view in greater detail.”
Link Floer homology and the Thurston norm (2006, background): “ts, such as Seiberg-Witten or Donaldson’s theories (compare [2]). In a diﬀerent direction, there seems to be a close connection between knot Floer homology and Khovanov’s homology for links (cf. [4], [5], [1], [7]), see also recent work of Rasmussen [22]. It is an open problem whether Khovanov’s hom”
Fixing the functoriality of Khovanov homology (2007, methodology): “In [16], Khovanov and Rozansky define a link homology theory which is a categorification of the generalized Jones polynomial based on Uq.”
A slice genus lower bound from sl(n) Khovanov–Rozansky homology (2007, methodology): “ in CKhnlies in the 0th chain group summand H(Bo). Up to non-zero scalar multiplication: α˜ξ= xn−1 1 x n−1 2 x n−1 3 ∈ (C[x1,x2,x3]/x n 1x n 2x n 3){3 −3n} = H(Bo) 60 Unpacking the proofs a little in [5], we see that (for the Reidemeister II.1 move which reduces the number of crossings and for the R”
Matrix factorizations and Cohomological Field Theories (2011, positive): “k and tensorproduct functors deﬁned above we obtaintheexternal tensor product functor MF(X,W)⊗MF(X′,W′) → MF(U,p∗ 1W+p ∗ 2W). (1.9) We can deﬁne Koszul matrix factorizations {α,β} in our setting (see [36], [54]). 9 Deﬁnition 1.1.8. Let X, Land Wbe as above, and let V be a vector bundle V on X. For g”
A category of kernels for equivariant factorizations and its implications for Hodge theory (2011, background): “dge theory associated to the category of matrix factorizations. Forthe case of an isolated local hypersurface singularity, T. Dyckerhoﬀ proved, in [Dyc09], that the category of kernels, introduced in [KR08a], is the correct one from the perspective of [To¨e07]. More precisely, the dg-category of ker”
Grid Homology for Knots and Links (2015, background): “It is interesting to compare the formal structure of knot Floer homology with a different kind of knot invariant defined by Mikhail Khovanov [103], and its various generalizations due to Khovanov and Lev Rozansky [104, 105].”
The universal sl3–link homology (2006, methodology): “of relations, one of which is X3 = 0. Gornik [4] studied the case when X3 = 1, which is the analogue of Lee’s work for sl3. To be precise, Gornik studied a deformation of the Khovanov-Rozansky theory [9] for sln, where nis arbitrary. Khovanov and Rozansky followed a different approach to link homolo”
DAHA and iterated torus knots (2014, background): “2], are missing (by now) upon the switch from the HOMFLYPT polynomials to their homology from [27; 28; 26; 48; 39] (generalizing the Khovanov homology for the Jones polynomials).”