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 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 . Ils sont étroitement liés à l’approche de Deligne-Gross des systèmes de racines exceptionnels ; ce thème est de nature expérimentale.
@article{AFST_2016_6_25_2-3_433_0,
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/}
}
TY - JOUR AU - Ivan Cherednik AU - Ross Elliot TI - Refined composite invariants of torus knots via DAHA JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2016 SP - 433 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 ER -
%0 Journal Article %A Ivan Cherednik %A Ross Elliot %T Refined composite invariants of torus knots via DAHA %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2016 %P 433-471 %V 25 %N 2-3 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1501/ %R 10.5802/afst.1501 %G en %F AFST_2016_6_25_2-3_433_0
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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