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DAHA-Jones polynomials of torus knots
- Publication Year :
- 2014
-
Abstract
- DAHA-Jones polynomials of torus knots T(r, s) are studied systematically for reduced root systems and in the case of $$C^\vee C_1$$ . We prove the polynomiality and evaluation conjectures from the author’s previous paper on torus knots and extend the theory by the color exchange and further symmetries. The DAHA-Jones polynomials for $$C^\vee C_1$$ depend on five parameters. Their surprising connection to the DAHA-superpolynomials (type A) for the knots $$T(2p+1,2)$$ is obtained, a remarkable combination of the color exchange conditions and the author’s duality conjecture (justified by Gorsky and Negut). The uncolored DAHA-superpolynomials of torus knots are expected to coincide with the Khovanov–Rozansky stable polynomials and the superpolynomials defined via rational DAHA and/or in terms of certain Hilbert schemes. We end the paper with certain arithmetic counterparts of DAHA-Jones polynomials for the absolute Galois group in the case of $$C^\vee C_1$$ , developing the author’s previous results for $$A_1$$ .
- Subjects :
- General Mathematics
General Physics and Astronomy
Duality (optimization)
Jones polynomial
Type (model theory)
01 natural sciences
Torus knot
Combinatorics
Mathematics::Quantum Algebra
0103 physical sciences
Mathematics - Quantum Algebra
FOS: Mathematics
Quantum Algebra (math.QA)
Representation Theory (math.RT)
0101 mathematics
Connection (algebraic framework)
Mathematics::Representation Theory
Mathematics::Symplectic Geometry
Mathematics
Conjecture
010102 general mathematics
Torus
Absolute Galois group
Mathematics::Geometric Topology
Algebra
010307 mathematical physics
Mathematics - Representation Theory
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....ad404d6e594b58eba7d197baeaf7502d