HOMFLYPT homology for links in handlebodies via type A Soergel bimodules

@article{Rose2019HOMFLYPTHF,
  title={HOMFLYPT homology for links in handlebodies via type A Soergel bimodules},
  author={David E. V. Rose and Daniel Tubbenhauer},
  journal={arXiv: Quantum Algebra},
  year={2019}
}
We define a triply-graded invariant of links in a genus g handlebody, generalizing the colored HOMFLYPT (co)homology of links in the 3-ball. Our main tools are the description of these links in terms of a subgroup of the classical braid group, and a family of categorical actions built from complexes of (singular) Soergel bimodules. 
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References

SHOWING 1-10 OF 48 REFERENCES
Triply-graded link homology and Hochschild homology of Soergel bimodules
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…
A geometric construction of colored HOMFLYPT homology
The aim of this paper is two-fold. First, we give a fully geometric description of the HOMFLYPT homology of Khovanov-Rozansky. Our method is to construct this invariant in terms of the cohomology of…
Gaitsgory's central sheaves via the diagrammatic Hecke category
We initiate the study of Gaitsgory's central sheaves using complexes of Soergel bimodules, in extended affine type A. We conjecture that the complex associated to the standard representation can be…
Categorification of the Kauffman bracket skein module of I-bundles over surfaces
Khovanov defined graded homology groups for links LR 3 and showed that their polynomial Euler characteristic is the Jones polyno- mial of L. Khovanov's construction does not extend in a…
A colored sl(N)-homology for links in S^3
  • Hao Wu
  • Mathematics, Biology
  • 2009
TLDR
It is proved that the homotopy type of this chain complex is invariant under Reidemeister moves and decategorifies to the Reshetikhin-Turaev sl(N) polynomial of links colored by exterior powers of the defining representation.
Khovanov-Rozansky homology and 2-braid groups
Khovanov has given a construction of the Khovanov-Rozansky link invariants (categorifying the HOMFLYPT invariant) using Hochschild cohomology of 2-braid groups. We give a direct proof that his…
Matrix factorizations and link homology
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…
Sutured annular Khovanov-Rozansky homology
We introduce an sl(n) homology theory for knots and links in the thickened annulus. To do so, we first give a fresh perspective on sutured annular Khovanov homology, showing that its definition…
Quadruply-graded colored homology of knots
We conjecture the existence of four independent gradings in colored HOMFLYPT homology, and make qualitative predictions of various interesting structures and symmetries in the colored homology of…
Categorification of sl 2 and braid groups
We describe categorifications of sl2 and braid groups. In a first part, we give a survey of the case of sl2 (joint work with Joseph Chuang [ChRou]) and explain how it leads to the construction of…
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