Jones polynomials of torus knots via DAHA

By I. Cherednik. 2011

155 citations recorded by the source corpus.

We suggest a construction for the Quantum Groups–Jones polynomials of torus knots in terms of the PBW theorem of double affine Hecke algebra (DAHA) for any root systems and weights (justified for type A). The main focus is on the DAHA super-polynomials, a stable type A variant of this construction. A connection is expected with the approach to super-polynomials due to Aganagic and Shakirov via the Macdonald polynomials at roots of unity and the Verlinde algebra. The duality conjecture for the DAHA super-polynomials is stated, essentially matching that due to Gukov and Stosic. A link to Khovanov–Rozansky polynomials is provided. The hyper-polynomials of types B and C are defined, generalizing the Kauffman polynomials. The special values and other features of the DAHA super- and hyper-polynomials are discussed.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/469d1cc1b6cadd2a6f223832d172a221e3061e26

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In Discrete Mathematics and Combinatorics

1. Andrei Neguct
2. E. Gorsky
3. Sergei Gukov
4. Jacob Rasmussen
5. Pavel Putrov
6. I. Danilenko
7. C. Vafa
8. I. Cherednik

In Geometry and Topology

1. Sergei Gukov
2. Pavel Putrov
3. I. Cherednik
4. E. Gorsky
5. C. Vafa
6. Jacob Rasmussen
7. Andrei Neguct
8. I. Danilenko

Citations in the corpus

• BPS spectra and 3-manifold invariants (2017, methodology): “In this section, we wish to compare the categorification of WRT invariants of knots in 3- manifolds considered in the previous section with analogous calculations using the tools of refined CS theory and DAHA [19, 72].”
• On dualizability of braided tensor categories (2018, background): “…should provide a foundational framework for: quantum A-polynomials [Dim13, Gar04, Guk05, GS12], DAHAJones polynomials and skein modules of knots complements [Che13, BS16, Sam17], and refined Chern–Simons invariants [AS15], among others, each of which yet lacks a topologically invariant formulation.”
• Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology (2016, methodology): “[17] I. Cherednik.”
• DAHA and iterated torus knots (2014, methodology): “(v) Superduality [25; 12; 22] Up to a power of q and t , (2-23) HEr;Es.”
• Refined knot invariants and Hilbert schemes (2013, methodology): “We use this viewpoint to pr ove two conjectures announced by Cherednik in [10] (let us remark that while the present paper was being written, Cherednik also announced independent proofs in [10, Section 2.”
• Fivebranes and 3-manifold homology (2016, background): “On the other hand, grading by U(1)β is easier to implement in the path integral and was heavily explored in [49, 57, 58].”