scholarly journals Exceptional knot homology

2016 ◽  
Vol 25 (03) ◽  
pp. 1640003
Author(s):  
Ross Elliot ◽  
Sergei Gukov
Keyword(s):  
Hecke Algebras ◽  
Torus Knots ◽  
Knot Invariants ◽  
Exceptional Groups ◽  
Double Affine Hecke Algebras ◽  
Natural Home ◽  
Knot Homology ◽  
Affine Hecke Algebras ◽  
Bps States ◽  
Knots And Links

The goal of this paper is twofold. First, we find a natural home for the double affine Hecke algebras (DAHA) in the physics of BPS states. Second, we introduce new invariants of torus knots and links called hyperpolynomials that address the “problem of negative coefficients” often encountered in DAHA-based approaches to homological invariants of torus knots and links. Furthermore, from the physics of BPS states and the spectra of singularities associated with Landau–Ginzburg potentials, we also describe a rich structure of differentials that act on homological knot invariants for exceptional groups and uniquely determine the latter for torus knots.

scholarly journals Double affine Hecke algebras and generalized Jones polynomials

2016 ◽  
Vol 152 (7) ◽  
pp. 1333-1384 ◽  
Author(s):  
Yuri Berest ◽  
Peter Samuelson
Keyword(s):  
Torus Knots ◽  
Knot Invariants ◽  
Skein Module ◽  
Double Affine Hecke Algebra ◽  
Double Affine Hecke Algebras ◽  
Jones Polynomials ◽  
Kauffman Bracket Skein Module ◽  
Affine Hecke Algebras ◽  
Figure Eight Knot ◽  
Colored Jones Polynomials

In this paper we propose and discuss implications of a general conjecture that there is a natural action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K\subset S^{3}$. We prove this in a number of nontrivial cases, including all $(2,2p+1)$ torus knots, the figure eight knot, and all 2-bridge knots (when $q=\pm 1$). As the main application of the conjecture, we construct three-variable polynomial knot invariants that specialize to the classical colored Jones polynomials introduced by Reshetikhin and Turaev. We also deduce some new properties of the classical Jones polynomials and prove that these hold for all knots (independently of the conjecture). We furthermore conjecture that the skein module of the unknot is a submodule of the skein module of an arbitrary knot. We confirm this for the same example knots, and we show that this implies that the colored Jones polynomials of $K$ satisfy an inhomogeneous recursion relation.

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