DAHA and skein algebra of surfaces: double-torus knots

  • Kazuhiro HikamiEmail author


We study a topological aspect of rank-1 double-affine Hecke algebra (DAHA). Clarified is a relationship between the DAHA of \(A_1\)-type (resp. \(C^\vee C_1\)-type) and the skein algebra on a once-punctured torus (resp. a 4-punctured sphere), and the \(SL(2;\mathbb {Z})\) actions of DAHAs are identified with the Dehn twists on the surfaces. Combining these two types of DAHA, we construct the DAHA representation for the skein algebra on a genus-two surface, and we propose a DAHA polynomial for a double-torus knot, which is a simple closed curve on a genus-two Heegaard surface in \(S^3\). Discussed is a relationship between the DAHA polynomial and the colored Jones polynomial.


Knot Colored Jones polynomial Double-affine Hecke algebra Skein algebra Macdonald polynomial Askey–Wilson polynomial 

Mathematics Subject Classification

57M27 57M25 20C08 33D52 57M60 81R12 



The author would like to thank H. Fuji, A.N. Kirillov, and H. Murakami for communications and comments on a draft of the manuscript. A part of this work was presented at the workshop “Volume Conjecture in Tokyo” on August 2018, celebrating the 60th birthday of Jun Murakami and Hitoshi Murakami. Thanks to the organizers and the participants. This work is supported in part by KAKENHI JP16H03927, JP17K05239, JP17K18781.


  1. 1.
    Aganagic, M., Shakirov, S.: Refined Chern–Simons theory and knot homology. In: Block, J., Distler, J., Donagi, R., Sharpe, E (eds.) String-Math 2011, Proceedings of Symposia in Pure Mathematics, vol. 85, pp. 3–31. AMS, Providence(2012)Google Scholar
  2. 2.
    Arthamonov, S., Shakirov, S.: Genus two generalization of \(A_1\) spherical DAHA, preprint (2017) arXiv:1704.02947v1 [math.QA]
  3. 3.
    Askey, R., Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Am. Math. Soc. 54, 1–55 (1985)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bakshi, R.P., Mukherjee, S., Przytycki, J.H., Silvero, M., Wang, X.: On multiplying curves in the Kauffman bracket skein algebra of the thickened four-holed sphere, preprint (2018). arXiv:1805.06062 [math.GT]
  5. 5.
    Berest, Y., Samuelson, P.: Double affine Hecke algebras and generalized Jones polynomials. Compos. Math. 152, 1333–1384 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Berest, Y., Samuelson, P.: Affine cubic surfaces and character varieties of knots. J. Algebra 500, 644–690 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Birman, J.S.: Braids, Links, and Mapping Class Groups. Princeton Univ. Press, Princeton (1974)Google Scholar
  8. 8.
    Bullock, D.: Rings of \(SL_2(\mathbb{C})\)-characters and the Kauffman bracket skein module. Comment. Math. Helv. 72, 521–542 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bullock, D., Przytycki, J.H.: Multiplicative structure of Kauffman bracket skein module quantization. Proc. Am. Math. Soc. 128, 923–931 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cha, J.C., Livingston, C.: KnotInfo: Table of knot invariants. Accessed 16 Nov 2018
  11. 11.
    Champanerkar, A., Kofman, I., Patterson, E.: The next simplest hyperbolic knots. J. Knot Theory Ramif. 13, 965–987 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cherednik, I.: Double Affine Hecke Algebras. Cambridge Univ. Press, Cambridge (2005)CrossRefzbMATHGoogle Scholar
  13. 13.
    Cherednik, I.: Jones polynomials of torus knots via DAHA. Int. Math. Res. Not. 2013, 5366–5425 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cherednik, I.: DAHA-Jones polynomials of torus knots. Selecta Math. (N.S.) 22, 1013–1053 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cho, J.-P., Gelca, R.: Representations of the Kauffman bracket skein algebra of the punctured torus. Fund. Math. 225, 45–55 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Culler, M., Dunfield, N.M., Goerner, M., Weeks, J.R.: SnapPy, a computer program for studying the geometry and topology of \(2\)-manifolds. Accessed 29 Nov 2018
  17. 17.
    Di Francesco, P., Kedem, R.: (\(t,q\))-deformed q-systems, DAHA and quantum toroidal algebras via generalized Macdonald operators, preprint (2017). arXiv:1704.00154 [math-ph]
  18. 18.
    Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton Univ. Press, Princeton (2011)CrossRefzbMATHGoogle Scholar
  19. 19.
    Frohman, C., Gelca, R.: Skein modules and the noncommutative torus. Trans. Am. Math. Soc. 352, 4877–4888 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Fuji, H., Gukov, S., Sułkowski, P.: Super-A-polynomial for knots and BPS states. Nucl. Phys. B 867, 506–546 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gasper, G., Rahman, M.: Basic Hypergeometric Series, 2nd edn. Cambridge Univ. Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  22. 22.
    Gukov, S., Nawata, S., Saberi, I., Stošić, M., Sułkowski, P.: Sequencing BPS spectra. J. High Energy Phys. 1603, 004–160 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Habiro, K.: A unified Witten–Reshetikhin–Turaev invariant for integral homology sphere. Invent. Math. 171, 1–81 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hikami, K.: Dunkl operator formalism for quantum many-body problems associated with the classical root systems. J. Phys. Soc. Jpn. 65, 394–401 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hikami, K.: Note on character varieties and cluster algebras. SIGMA 15, 003 (2019). 32 pagesMathSciNetzbMATHGoogle Scholar
  26. 26.
    Hikami, K.: Work in progressGoogle Scholar
  27. 27.
    Hill, P.: On double-torus knots I. J. Knot Theory Ramif. 8, 1009–1048 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hill, P., Murasugi, K.: On double-torus knots II. J. Knot Theory Ramif. 9, 617–667 (2000)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Kirillov, A.N., Noumi, M.: Affine Hecke algebras and raising operators for Macdonald polynomials. Duke Math. J. 93, 1–39 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Koornwinder, T.H.: Lowering and raising operators for some special orthogonal polynomials. In: Kuznetsov, V.B., Sahi, S. (eds.) Jack, Hall–Littlewood and Macdonald Polynomials, vol. 417 of Contemporary Mathematics, pp. 227–238, Amer. Math. Soc., Providence (2006)Google Scholar
  31. 31.
    Koornwinder, T.H.: The relationship between Zhedanov’s algebra AW(3) and the double affine Hecke algebra in the rank one case. SIGMA 3, 15–63 (2007)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Koornwinder, T.H.: Zhedanov’s algebra AW(3) and the double affine Hecke algebra in the rank one case II. The spherical subalgebra. SIGMA 4, 052 (2008). 17 pagesMathSciNetzbMATHGoogle Scholar
  33. 33.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford Univ. Press, Oxford (1995)zbMATHGoogle Scholar
  34. 34.
    Macdonald, I.G.: Affine Hecke Algebras and Orthogonal Polynomials. Cambridge Univ. Press, Cambrdige (2003)CrossRefzbMATHGoogle Scholar
  35. 35.
    Marché, J., Paul, T.: Toeplitz operators in TQFT via skein theory. Trans. Am. Math. Soc. 367, 3669–3704 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Masbaum, G.: Skein-theoretical derivation of some formulas of Habiro. Algebraic Geom Topol 3, 537–556 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Morton, H.R.: The coloured Jones function and Alexander polynomial for torus knots. Proc. Camb. Philos. Soc. 117, 129–135 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Noumi, M.: Macdonald–Koornwinder polynomial and affine Hecke algebra. RIMS Kôkyûroku 919, 44–55 (1995). in JapanesezbMATHGoogle Scholar
  39. 39.
    Noumi, M., Stokman, J.V.: Askey–Wilson polynomials: an affine Hecke algebraic approach. In: Alvarez-Nodarse, R., Marcellan, F., van Assche, W. (eds.) Laredo Lectures on Orthogonal Polynomials and Special Functions, pp. 111–144. Nova Science Pub, New York (2004)Google Scholar
  40. 40.
    Oblomkov, A.: Double affine Hecke algebras of rank \(1\) and affine cubic surfaces. IMRN 2004, 877–912 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Przytycki, J.H., Sikora, A.S.: On skein algebra and Sl\({}_2(C)\)-character varieties. Topology 39, 115–148 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Sahi, S.: Nonsymmetric Koornwinder polynomials and duality. Ann. Math. 150, 267–282 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Sklyanin, E.K.: Boundary conditions for integrable quantum systems. J. Phys. A Math. Gen. 21, 2375–2389 (1988)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Terwilliger, P.: The universal Askey–Wilson algebra and DAHA of type \((C_1^\vee, C_1)\). SIGMA 9, 047 (2013). 40 pageszbMATHGoogle Scholar
  45. 45.
    The Knot Atlas. Accessed 24 Nov 2018
  46. 46.
    Zhedanov, A.S.: “Hidden symmetry” of Askey–Wilson polynomials. Theor. Math. Phys. 89, 1146–1157 (1991)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Faculty of MathematicsKyushu UniversityFukuokaJapan

Personalised recommendations