Matrix factorizations and link homology II

By Mikhail Khovanov, L. Rozansky. 2005

363 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/2a5e9df7fa8e02f1250dcd779b838642b60ccb0b

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

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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.”
A diagrammatic category for the representation theory of ( Uqsln ) (2007, methodology): “r sl 3 link homology categoriﬁes the U q(sl 3) theory, although it’s only made 73 explicitly local in later work by Ari Nieh and myself [25]. Finding an alternative to the matrix factorisation method [19, 20] of categorifying the U q(sl n) knot invariants, based explicitly on a categoriﬁcation of th”
q,t-Catalan numbers and knot homology (2010, background): “Later Khovanov and Rozansky gave a unified construction ([19]) of the homology theories categorifying sl(N) Jones polynomials, and also another homology theory ([20]) categorifying the HOMFLY polynomial.”
Heegaard Floer homology and integer surgeries on links (2010, neutral): “[6] M. Khovanov and L. Rozansky, Matrix factorizations and link homology II, Geom.”
The Superpolynomial for Knot Homologies (2005, background): “There are also important deformations of the original Khovanov homology [Lee 02b, Bar-Natan 05a, Khovanov 04b], as well as of the sl(N) Khovanov–Rozansky homology [Gornik 04].”
Holomorphic disks, link invariants and the multi-variable Alexander polynomial (2008, background): “me degree, link Floer homology can be viewed as a categoriﬁcation of the multivariable Alexander polynomial. It is interesting to compare this with the recent categoriﬁcation of the HOMFLY polynomial [10], see also [8], [9], [17]. Acknowledgements. We would like to thank Robert Lipshitz, Jacob Rasmu”