Skip to main content
SpringerLink
Log in

Cyclotomic expansion of generalized Jones polynomials

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

In (Compos. Math. 152(7): 1333–1384, 2016), Berest and Samuelson proposed a conjecture that the Kauffman bracket skein module of any knot in \(S^3\) carries a natural action of a rank 1 double-affine Hecke algebra \(SH_{q,t_1, t_2}\) depending on 3 parameters \(q, t_1, t_2\). As a consequence, for a knot K satisfying this conjecture, we defined a three-variable polynomial invariant \(J^K_n(q,t_1,t_2)\) generalizing the classical coloured Jones polynomials \(J^K_n(q)\). In this paper, we give explicit formulas and provide a quantum group interpretation for the polynomials \(J^K_n(q,t_1,t_2)\). Our formulas generalize the so-called cyclotomic expansion of the classical Jones polynomials constructed by Habiro (Invent. Math. 171(1): 1–81, 2008) : as in the classical case, they imply the integrality of \(J^K_n(q,t_1,t_2)\) and, in fact, make sense for an arbitrary knot K independent of whether or not it satisfies the conjecture of Berest and Samuelson (Compos. Math. 152(7): 1333–1384, 2016). When one of the Hecke deformation parameters is set to be 1, we show that the coefficients of the (generalized) cyclotomic expansion of \(J^K_n(q,t_1)\) are expressed in terms of Macdonald orthogonal polynomials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. By contrast, the polynomials \(J^K_n(q,t_1,t_2)\) cannot be written, in general, as linear combinations of the classical coloured Jones polynomials \(J_n^K(q)\) with coefficients in \({\mathbb {C}}[q^{\pm 1}, t_1^{\pm 1}, t_2^{\pm 1}]\) (cf. Remark 3.4 in Sect. 3.2).

  2. For this reason and also to avoid confusion with terminology of [1] we will keep referring to \(J^K_n(q,t_1,t_2)\) as ‘generalized’ Jones polynomials, even though they are not generalizing the classical Jones polynomials in the same way as other multivariable polynomial knot invariants (arising, for example, from Khovanov homology).

  3. Abusing notation, we denote the extended pairing of nonsymmetric skein modules in the same way as the ‘symmetric’ (topological) one.

  4. The polynomials \(C_n(x;\beta | q)\) are sometimes called the q-ultraspherical (or Rogers) polynomials (cf. [11, Sect. 14.10.1]).

  5. Lawrence’s universal invariants can be defined for more general Lie algebras than \(\mathfrak {sl}_2\) and for more general link-type diagrams (bottom tangles), see [7].

  6. We warn the reader that our q differs from the q in [8]: in fact, the q in [8] equals \(v^2\), which is our \(q^4\).

  7. An (m, n)-tangle is a properly embedded 1-manifold in \([0,1]\times [0,1]\) with m endpoints on \(\{0\}\times [0,1]\) and n endpoints on \(\{1\} \times [0,1]\).

References

  1. Berest, Y., Samuelson, P.: Double affine Hecke algebras and generalized Jones polynomials. Compos. Math. 152(7), 1333–1384 (2016)

    Article  MathSciNet  Google Scholar 

  2. Berest, Y., Samuelson, P.: Affine cubic surfaces and character varieties of knots. J. Algebra 500, 644–690 (2018)

    Article  MathSciNet  Google Scholar 

  3. Cherednik, Ivan: Double affine Hecke algebras, London Mathematical Society Lecture Note Series, vol. 319. Cambridge University Press, Cambridge (2005)

    Book  Google Scholar 

  4. Cautis, S., Kamnitzer, J., Morrison, S.: Webs and quantum skew Howe duality. Math. Ann. 360(1–2), 351–390 (2014)

    Article  MathSciNet  Google Scholar 

  5. Frohman, C., Gelca, R.: Skein modules and the noncommutative torus. Trans. Am. Math. Soc. 352(10), 4877–4888 (2000)

    Article  MathSciNet  Google Scholar 

  6. Garoufalidis, S.L., Thang, T.Q.: Asymptotics of the colored Jones function of a knot. Geom. Topol. 15(4), 2135–2180 (2011)

    Article  MathSciNet  Google Scholar 

  7. Habiro, Kazuo: Bottom tangles and universal invariants. Algebr. Geom. Topol. 6, 1113–1214 (2006)

    Article  MathSciNet  Google Scholar 

  8. Habiro, Kazuo: A unified Witten-Reshetikhin–Turaev invariant for integral homology spheres. Invent. Math. 171(1), 1–81 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  9. Kashaev, R.M.: The hyperbolic volume of knots from the quantum dilogarithm. Lett. Math. Phys. 39(3), 269–275 (1997)

    Article  MathSciNet  Google Scholar 

  10. Louis, H.: Kauffman, State models and the Jones polynomial. Topology 26(3), 395–407 (1987)

    Article  MathSciNet  Google Scholar 

  11. Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric orthogonal polynomials and their \(q\)-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, (2010), With a foreword by Tom H. Koornwinder

  12. Kirby, R., Melvin, P.: The \(3\)-manifold invariants of Witten and Reshetikhin-Turaev for \({\rm sl}(2,{ C})\). Invent. Math. 105(3), 473–545 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  13. Kuperberg, Greg: Spiders for rank \(2\) Lie algebras. Comm. Math. Phys. 180(1), 109–151 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  14. Lawrence, R.J.: A universal link invariant using quantum groups, Differential geometric methods in theoretical physics (Chester, 1988) : World Sci. Publ. Teaneck, NJ, pp. 55–63 (1989)

  15. Lawrence, R.J.: A universal link invariant, The interface of mathematics and particle physics (Oxford, : Inst. Math. Appl. Conf. Ser. New Ser.), vol. 24, Oxford University, Press, New York 1990, 151–156 (1988)

  16. Masbaum, Gregor: Skein-theoretical derivation of some formulas of Habiro. Algebr. Geom. Topol. 3, 537–556 (2003)

    Article  MathSciNet  Google Scholar 

  17. Murakami, H., Murakami, J.: The colored Jones polynomials and the simplicial volume of a knot. Acta Math. 186(1), 85–104 (2001)

    Article  MathSciNet  Google Scholar 

  18. Murakami, H.: An introduction to the volume conjecture. Interactions between hyperbolic geometry quantum topology and number theory Contemporary Mathematics. vol. 541, pp. 1–40. American Mathematical Society, Providence (2011)

  19. Noumi, M., Stokman, J.V.: Askey-Wilson polynomials: an affine Hecke algebra approach, Laredo Lectures on Orthogonal Polynomials and Special Functions, Adv. Theory Spec. Funct. Orthogonal Polynomials. pp. 111–144. Nova Science Publishers, Hauppauge, NY, (2004)

  20. Oblomkov, Alexei: Double affine Hecke algebras of rank 1 and affine cubic surfaces. Int. Math. Res. Not. 18, 877–912 (2004)

    Article  MathSciNet  Google Scholar 

  21. Józef, H.: Przytycki, Skein modules of \(3\)-manifolds. Bull. Polish Acad. Sci. Math. 39(1–2), 91–100 (1991)

    MathSciNet  MATH  Google Scholar 

  22. Reshetikhin, N.Y., Turaev, V.G.: Ribbon graphs and their invariants derived from quantum groups. Commun. Math. Phys. 127(1), 1–26 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  23. Sahi, S.: Nonsymmetric Koornwinder polynomials and duality. Ann. Math. 150(1), 267–282 (1999)

    Article  MathSciNet  Google Scholar 

  24. Sikora, Adam S., Westbury, Bruce W.: Confluence theory for graphs. Algebra Geom. Topol. 7, 439–478 (2007)

    Article  MathSciNet  Google Scholar 

  25. Tingley, P.: A minus sign that used to annoy me but now I know why it is there (two constructions of the Jones polynomial). In Proceedings of the 2014 Maui and 2015 Qinhuangdao conferences in honour of Vaughan F. R. Jones’ 60th birthday, Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 46, Austral. Nat. Univ., Canberra, pp. 415–427 (2017)

  26. Vladimir, G.: Turaev, Skein quantization of Poisson algebras of loops on surfaces. Ann. Sci. École Norm. Sup. 24(6), 635–704 (1991)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank I. Cherednik, P. Di Francesco, N. Reshetikhin, and V. Turaev for interesting discussions, questions, and comments. We are especially grateful to G. Felder who clarified to us the structure of cyclotomic coefficients and their role in the Volume Conjecture. The work of the first author (Yu. B.) was partially supported by the NSF grant DMS 1702372 and the 2019 Simons Fellowship both of which are gratefully acknowledged. The work of the third author was partially supported by a Simons Travel Grant and a Simons Collaboration Grant which are also gratefully acknowledged.

Author information

Authors and Affiliations

  1. Department of Mathematics, Cornell University, Ithaca, USA

    Yuri Berest & Joseph Gallagher

  2. Department of Mathematics, University of California, Riverside, USA

    Peter Samuelson

Authors
  1. Yuri Berest
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joseph Gallagher
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Peter Samuelson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuri Berest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Berest, Y., Gallagher, J. & Samuelson, P. Cyclotomic expansion of generalized Jones polynomials. Lett Math Phys 111, 37 (2021). https://doi.org/10.1007/s11005-021-01373-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11005-021-01373-6

Keywords

Mathematics Subject Classification

Search

Navigation