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, 25:58 | Cite as

Cluster algebras and Jones polynomials

  • Kyungyong Lee
  • Ralf SchifflerEmail author
Article
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Abstract

We present a new and very concrete connection between cluster algebras and knot theory. This connection is being made via continued fractions and snake graphs. It is known that the class of 2-bridge knots and links is parametrized by continued fractions, and it has recently been shown that one can associate to each continued fraction a snake graph, and hence a cluster variable in a cluster algebra. We show that up to normalization by the leading term the Jones polynomial of the 2-bridge link is equal to the specialization of this cluster variable obtained by setting all initial cluster variables to 1 and specializing the initial principal coefficients of the cluster algebra as follows \(y_1=t^{-2}\) and \( y_i=-t^{-1}\), for all \(i> 1\). As a consequence we obtain a direct formula for the Jones polynomial of a 2-bridge link as the numerator of a continued fraction of Laurent polynomials in \(q=-t^{-1}\). We also obtain formulas for the degree and the width of the Jones polynomial, as well as for the first three and the last three coefficients. Along the way, we also develop some basic facts about even continued fractions and construct their snake graphs. We show that the snake graph of an even continued fraction is isomorphic to the snake graph of a positive continued fraction if the continued fractions have the same value. We also give recursive formulas for the Jones polynomials.

Keywords

Cluster algebras Jones polynomial 2-Bridge knots Continued fractions Snake graphs 

Mathematics Subject Classification

Primary 13F60 Secondary 57M27 11A55 

Notes

References

  1. 1.
    Çanakçı, İ., Schiffler, R.: Snake graph calculus and cluster algebras from surfaces. J. Algebra 382, 240–281 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Çanakçı, İ., Schiffler, R.: Snake graph calculus and cluster algebras from surfaces II: self-crossing snake graphs. Math. Z. 281(1), 55–102 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Çanakçı, İ., Schiffler, R.: Snake graph calculus and cluster algebras from surfaces III: band graphs and snake rings. Int. Math. Res. Not. 2019, 1145–1226 (2017)CrossRefGoogle Scholar
  4. 4.
    Çanakçı, İ., Schiffler, R.: Cluster algebras and continued fractions. Compos. Math. 154(3), 565–593 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Conway, J.H.: An enumeration of knots and links, and some of their algebraic properties. In: Leech, J. (ed.) Computational Problems in Abstract Algebra, pp. 329–358. Pergamon Press, Oxford (1970)Google Scholar
  6. 6.
    Derksen, H., Weyman, J., Zelevinsky, A.: Quivers with potentials and their representations II: applications to cluster algebras. J. Am. Math. Soc. 23(3), 749–790 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fomin, S., Shapiro, M., Thurston, D.: Cluster algebras and triangulated surfaces, Part I: cluster complexes. Acta Math. 201, 83–146 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fomin, S., Zelevinsky, A.: Cluster algebras I: foundations. J. Am. Math. Soc. 15, 497–529 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fomin, S., Zelevinsky, A.: Cluster algebras IV: coefficients. Compos. Math. 143, 112–164 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Futer, D., Kalfagianni, E., Purcell, J.: Dehn filling, volume, and the Jones polynomial. J. Differ. Geom. 78(3), 429–464 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, fourth edn. Clarendon Press, Oxford (1960)zbMATHGoogle Scholar
  12. 12.
    Hikami, K., Inoue, R.: Braiding operator via quantum cluster algebra. J. Phys. A 47(47), 474006 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hikami, K., Inoue, R.: Braids, complex volume and cluster algebras. Algebr. Geom. Topol. 15(4), 2175–2194 (2015). (English summary)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jones, V.: A polynomial invariant for knots via von Neumann algebras. Bull. Am. Math. Soc. 12, 103–111 (1984)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kanenobu, T., Miyazawa, Y.: 2-bridge link projections. Kobe J. Math. 9(2), 171–182 (1992)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kauffman, L.H., Lambropoulou, S.: On the classification of rational knots. Enseign. Math. (2) 49(3–4), 357–410 (2003). (English summary)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Lee, K., Schiffler, R.: Positivity for cluster algebras. Ann. Math. 182(1), 73–125 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lickorish, W.: An Introduction to Knot Theory. Graduate Texts in Mathematics, vol. 175. Springer, New York (1997)CrossRefGoogle Scholar
  19. 19.
    Lickorish, W., Millet, K.: A polynomial invariant of oriented links. Topology 26(1), 107–141 (1987)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Musiker, G., Schiffler, R.: Cluster expansion formulas and perfect matchings. J. Algebr. Combin. 32(2), 187–209 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Musiker, G., Schiffler, R., Williams, L.: Positivity for cluster algebras from surfaces. Adv. Math. 227, 2241–2308 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Musiker, G., Schiffler, R., Williams, L.: Bases for cluster algebras from surfaces. Compos. Math. 149(2), 217–263 (2013)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Nakabo, S.: Formulas on the HOMFLY and Jones polynomials of 2-bridge knots and links. Kobe J. Math. 17(2), 131–144 (2000)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Perron, O.: Die Lehre von den Kettenbrüchen, 2nd edition, Chelsea Publishing Company, New York, 1950. (the first edition of this book from 1913 is available at https://archive.org/details/dielehrevondenk00perrgoog. Accessed 15 Jan 2017)
  25. 25.
    Propp, J.: The Combinatorics of Frieze Patterns and Markoff Numbers arXiv:math.CO/0511633
  26. 26.
    Rabideau, M.: \(F\)-polynomial formula from continued fractions. J. Algebra 509, 467–475 (2018)Google Scholar
  27. 27.
    Schiffler, R.: Homological Methods, Representation Theory, and Cluster Algebras. Lecture Notes on Cluster Algebras from Surfaces, pp. 65–99. CRM Short Courses, Springer, New York (2018)CrossRefGoogle Scholar
  28. 28.
    Schubert, H.: Knoten mit zwei Brücken. Math. Z. 65, 133–170 (1956). (German)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Shende, V., Treumann, D., Williams, H.: Cluster Varieties from Legendrian Knots, arXiv:1512.08942

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AlabamaTuscaloosaUSA
  2. 2.Korea Institute for Advanced StudySeoulRepublic of Korea
  3. 3.Department of MathematicsUniversity of ConnecticutStorrsUSA

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