Triply-graded link homology and Hochschild homology of Soergel bimodules

By Mikhail Khovanov. 2005

229 citations recorded by the source corpus.

We consider a class of bimodules over polynomial algebras which were originally introduced by Soergel in relation to the Kazhdan–Lusztig theory, and which describe a direct summand of the category of Harish–Chandra modules for sl(n). Rouquier used Soergel bimodules to construct a braid group action on the homotopy category of complexes of modules over a polynomial algebra. We apply Hochschild homology to Rouquier's complexes and produce triply-graded homology groups associated to a braid. These groups turn out to be isomorphic to the groups previously defined by Lev Rozansky and the author, which depend, up to isomorphism and overall shift, only on the closure of the braid. Consequently, our construction produces a homology theory for links.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d841b9d205ae5169afc30782df3bd92a289bbfee

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In Geometry and Topology

1. Jacob Rasmussen
2. E. Gorsky
3. I. Cherednik
4. Marko Stosic
5. Andrei Neguct
6. Sergei Gukov
7. I. Danilenko
8. Mikhail Khovanov

In Mathematical Physics

1. I. Cherednik
2. Jacob Rasmussen
3. Marko Stosic
4. E. Gorsky
5. Mikhail Khovanov
6. Andrei Neguct
7. Pedro Vaz
8. Sergei Gukov

Citations in the corpus

• A diagrammatic categorification of the q-Schur algebra (2010, background): “of Soergel’s monoidal category, which also categorifies the Hecke algebra Hq(n). For more history on Soergel’s work and categorification, se e the introduction to their paper. Also for g= sln, Khovanov [10] proved that the Hochschild homology 1 of Rouquier’s complex gives the Khovanov-Rozansky-HOMFLY-”
• The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link (2012, background): “In the .2; n/ case, these formulas match the known value of P [44]; in the .3; n/ case, the resulting formula matches a conjectural formula for P advanced in [20].”
• Some differentials on Khovanov–Rozansky homology (2006, methodology): “a KR-thin knot and a KR-thin link is also KR-thin. 7.5. Small knots. We conclude by describing the KR-homology of knots with 9 or fewer crossings. Previous computations of Hhave been made by Khovanov [10], who showed that the (2,n) torus knots are KR-thin, and by Webster [24], who wrote a computer p”
• 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.”
• 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).”
• Categorifications from planar diagrammatics (2010, methodology): “(c) Taking Hochschild homology of Rouquier complexes in a suitable way leads to homology groups that turn out to depend only on the closure of a braid [25].”
• Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology (2016, background): “Khovanov [39] defined the HOMFLY-PT homology of a braid σ as: (3.”
• Categorification of (induced) cell modules and the rough structure of generalized Verma modules (2007, unknown): “This statement can be found for example in [79] and [41].”
• Refined knot invariants and Hilbert schemes (2013, methodology): “e Aganagic-Shakirov and OblomkovRasmussen-Shende conjectural descriptions of HOMFLY homology of torus knots, though, indeed, the relation between both of these descriptions to the actual definition of [47, 46] remains unknown. Another conjecture relates the operator Pm,n to the finite-dimensional repr”
• Singular Soergel bimodules (2010, unknown): “rovide a means of avoiding the use of the decomposition theorem. This hope has not yet be realised.) (2) In [17] Khovanov and Rozansky constructed a categorification of the HOMFLYPT polyonomial and in [16] Khovanov gave another construction of this invariant by taking the Hochschild homology of a com”
• Braid cobordisms, triangulated categories, and flag varieties (2006, background): “ products carry an enormous amount of information, can be used to reconstruct a regular block of the category O for sln,see [So], and serve as building blocks for a triply-graded link homology theory [Kh5]. Remark: Since we can work over Zrather than over k, the projectivity is at most in the minus ”