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Refined composite invariants of torus knots via DAHA
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 2-3, pp. 433-471.

We define composite DAHA-superpolynomials of torus knots, depending on pairs of Young diagrams and generalizing the composite HOMFLY-PT polynomials in the skein theory of the annulus. We provide various examples. Our superpolynomials extend the DAHA-Jones (refined) polynomials and satisfy all standard symmetries of the DAHA-superpolynomials of torus knots. The latter are conjecturally related to the HOMFLY-PT homology. At the end, we construct two DAHA-hyperpolynomials extending the DAHA-Jones polynomials of type E closely related to the Deligne-Gross approach to the exceptional root systems; this theme is of experimental nature.

Nous définissons les DAHA-superpolynômes composites associés aux nœuds toriques, en fonction des paires de diagrammes de Young qui généralisent les polynômes de HOMFLY-PT composites dans la théorie de skein de l’anneau. Nous donnons divers exemples. Nos superpolynômes étendent les polynômes (raffinés) de DAHA-Jones et satisfont toutes les symétries standards des DAHA-superpolynômes des nœuds toriques. Ces derniers sont conjecturalement liés à l’homologie de HOMFLY-PT. À la fin, nous construisons deux DAHA-hyperpolynômes en étendant les polynômes de DAHA-Jones de type E. Ils sont étroitement liés à l’approche de Deligne-Gross des systèmes de racines exceptionnels ; ce thème est de nature expérimentale.

Published online:
DOI: 10.5802/afst.1501
Ivan Cherednik 1; Ross Elliot 2

1 Department of Mathematics, UNC Chapel Hill, North Carolina 27599, USA
2 California Institute of Technology, Pasadena, California 91125, USA
     author = {Ivan Cherednik and Ross Elliot},
     title = {Refined composite invariants of torus knots via {DAHA}},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {433--471},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 25},
     number = {2-3},
     year = {2016},
     doi = {10.5802/afst.1501},
     zbl = {1379.57008},
     mrnumber = {3530165},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1501/}
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JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
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EP  - 471
VL  - 25
IS  - 2-3
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1501/
DO  - 10.5802/afst.1501
LA  - en
ID  - AFST_2016_6_25_2-3_433_0
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Ivan Cherednik; Ross Elliot. Refined composite invariants of torus knots via DAHA. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 2-3, pp. 433-471. doi : 10.5802/afst.1501. https://afst.centre-mersenne.org/articles/10.5802/afst.1501/

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