Soergel bimodules and matrix factorizations

Abstract

We establish an isomorphism between the Khovanov-Rozansky triply graded link homology and the geometric triply graded homology due to the authors. Hence we provide an interpretation of the Khovanov-Rozansky homology of the closure of a braid $\beta$ as the space of derived sections of a ${\mathbb{C}}^{\ast }\times {\mathbb{C}}^{\ast }$- equivariant sheaf $Tr(\beta )$ on the Hilbert scheme $Hil{b}_n({\mathbb{C}}^2)$, thus proving a version of Gorsky-Negut-Rasmussen conjecture \cite{GorskyNegutRasmussen16}. As a consequence we prove that Khovanov-Rozansky homology of knots satisfies the $q\to t/q$ symmetry conjectured by Dunfield-Gukov-Rasmussen \cite{DunfieldGukovRasmussen06}. We also apply our main result to compute the Khovanov-Rozansky homology of torus links.

arXiv comment51 pages, 1 figure, comments are welcome

References in corpus (10)

• Triangulated categories of singularities and D-branes in Landau-Ginzburg models
• Vanishing theorems and character formulas for the Hilbert scheme of points in the plane
• The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link
• Koszul duality and modular representations of semi-simple Lie algebras
• Homology of torus knots
• Khovanov-Rozansky homology and higher Catalan sequences
• q,t-Catalan numbers and knot homology
• Braid group actions on matrix factorizations
• Dualizable link homology
• Equivariant Matrix Factorizations and Hamiltonian reduction

Cited by in corpus (2)

• Tautological classes and symmetry in Khovanov-Rozansky homology
• Algebra and geometry of link homology