Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-28T08:52:32.389Z Has data issue: false hasContentIssue false
  • English
  • Français

$q$-DEFORMED RATIONALS AND $q$-CONTINUED FRACTIONS

Part of: Arithmetic rings and other special rings Sequences and sets Elementary number theory Combinatorics Low-dimensional topology

Published online by Cambridge University Press:  06 March 2020

SOPHIE MORIER-GENOUD  and
VALENTIN OVSIENKO Open the ORCID record for VALENTIN OVSIENKO [Opens in a new window]
SOPHIE MORIER-GENOUD
Affiliation:
Sorbonne Université, Université Paris Diderot, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, F-75005, Paris, France; [email protected]
VALENTIN OVSIENKO
Affiliation:
Centre national de la recherche scientifique, Laboratoire de Mathématiques, UMR du CNRS 9008, U.F.R. Sciences Exactes et Naturelles Moulin de la Housse - BP 1039 51687 Reims cedex 2, France; [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We introduce a notion of $q$-deformed rational numbers and $q$-deformed continued fractions. A $q$-deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the $q$-deformed Pascal identity for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the $q$-rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties, $q$-deformation of the Farey graph, matrix presentations and $q$-continuants are given, as well as a relation to the Jones polynomial of rational knots.

MSC classification

Primary: 05A30: $q$-calculus and related topics 11A55: Continued fractions 11B57: Farey sequences; the sequences ${1^k, 2^k, cdots}$ 13F60: Cluster algebras 57M27: Invariants of knots and 3-manifolds
Type
Discrete Mathematics
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e13
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

References

Andrews, G., q-Series: their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra, CBMS Regional Conference Series in Mathematics, 66 (American Mathematical Society, Providence, RI, 1986).Google Scholar
Andrews, G., Baxter, R. and Forrester, P., ‘Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities’, J. Stat. Phys. 35 (1984), 193–266.CrossRefGoogle Scholar
Berstel, J. and Reutenauer, C., Noncommutative Rational Series with Applications, Encyclopedia of Mathematics and its Applications, 137 (Cambridge University Press, Cambridge, 2011).Google Scholar
Boca, F., ‘Products of matrices and and the distribution of reduced quadratic irrationals’, J. Reine Angew. Math. 606 (2007), 149–165.Google Scholar
Broline, D., Crowe, D. and Isaacs, I., ‘The geometry of frieze patterns’, Geom. Dedicata 3 (1974), 171–176.CrossRefGoogle Scholar
C˛anakc˛i, I. and Schiffler, R., ‘Snake graphs and continued fractions’, European J. Combin. 86 (2020), 103081.Google Scholar
Carlitz, L., ‘Fibonacci notes. III. q-Fibonacci numbers’, Fibonacci Quart. 12 (1974), 317–322.Google Scholar
Chekhov, L. and Penner, R., ‘On quantizing Teichmüller and Thurston theories’, inHandbook of Teichmüller theory, Vol. I (Mathematical Society, Zürich, 2007), 579–645.CrossRefGoogle Scholar
Conway, J. H. and Coxeter, H. S. M., ‘Triangulated polygons and frieze patterns’, Math. Gaz. 57 (1973), 87–94 and 175–183.CrossRefGoogle Scholar
Coxeter, H. S. M., ‘Frieze patterns’, Acta Arith. 18 (1971), 297–310.CrossRefGoogle Scholar
Derksen, H. and Weyman, J., An Introduction to Quiver Representations, Graduate Studies in Mathematics, 184 (American Mathematical Society, Providence, RI, 2017).CrossRefGoogle Scholar
Derksen, H., Weyman, J. and Zelevinsky, A., ‘Quivers with potentials and their representations II: applications to cluster algebras’, J. Amer. Math. Soc. 23(3) (2010), 749–790.CrossRefGoogle Scholar
Fok, V. and Chekhov, L., ‘Quantum Teichmüller spaces’, Theoret. Math. Phys. 120 (1999), 1245–1259.CrossRefGoogle Scholar
Fomin, S. and Zelevinsky, A., ‘Cluster algebras. IV. Coefficients’, Compos. Math. 143(1) (2007), 112–164.CrossRefGoogle Scholar
Graham, R., Knuth, D. and Patashnik, O., ‘Concrete mathematics’, inA foundation for computer science, Advanced Book Program (Addison-Wesley Publishing Company, Reading, MA, 1989), xiv+625 pp.Google Scholar
Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers, 6th edn. Revised by D. R. Heath-Brown and J. H. Silverman. With a foreword by Andrew Wiles, (Oxford University Press, Oxford, 2008), 621 pp.Google Scholar
Hirzebruch, F. E. P., ‘Hilbert modular surfaces’, Enseign. Math. (2) 19 (1973), 183–281.Google Scholar
Hirzebruch, F. and Zagier, D., ‘Classification of Hilbert modular surfaces’, inComplex Analysis and Algebraic Geometry, Collected Papers II (Iwanami Shoten, Tokyo, 1977), 43–77.CrossRefGoogle Scholar
Katok, S., ‘Coding of closed geodesics after Gauss and Morse’, Geom. Dedicata 63(2) (1996), 123–145.CrossRefGoogle Scholar
Kauffman, L. and Lambropoulou, S., ‘On the classification of rational knots’, Enseign. Math. (2) 49(3–4) (2003), 357–410.Google Scholar
Keller, B., Quiver mutation in Java, applet available at the author’s home page.Google Scholar
Khrabrov, A. and Kokhas, K., ‘Points on a line, shoelace and dominoes’, Preprint, 2015,arXiv:1505.06309.Google Scholar
Kogiso, T. and Wakui, M., ‘A Bridge between Conway-Coxeter Friezes and Rational Tangles through the Kauffman Bracket Polynomials’, Preprint, 2018, arXiv:1806.04840.CrossRefGoogle Scholar
Lee, K. and Schiffler, R., ‘Cluster algebras and Jones polynomials’, Selecta Math. (N.S.) 25(4) (2019), Art. 58, 41 pp.CrossRefGoogle Scholar
Morier-Genoud, S. and Ovsienko, V., ‘Farey boat I. Continued fractions and triangulations, modular group and polygon dissections’, Jahresber. Dtsch. Math.-Ver. 121(2) (2019), 91–136.CrossRefGoogle Scholar
Morier-Genoud, S. and Ovsienko, V., ‘On q-Deformed Real Numbers’, Exp. Math. doi:10.1080/10586458.2019.1671922, arXiv:1908.04365.Google Scholar
Munarini, E. and Zagaglia Salvi, N., ‘On the rank polynomial of the lattice of order ideals of fences and crowns’, Discrete Math. 259(1–3) (2002), 163–177.CrossRefGoogle Scholar
Ovsienko, V., ‘Partitions of unity in SL(2, ℤ), negative continued fractions, and dissections of polygons’, Res. Math. Sci. 5(2) (2018), Paper No. 21, 25 pp.CrossRefGoogle Scholar
Pan, H., ‘Arithmetic properties of q-Fibonacci numbers and q-Pell numbers’, Discrete Math. 306 (2006), 2118–2127.CrossRefGoogle Scholar
Rabideau, M., ‘F-polynomial formula from continued fractions’, J. Algebra 509 (2018), 467–475.CrossRefGoogle Scholar
Series, C., ‘The modular surface and continued fractions’, J. Lond. Math. Soc. (2) 31(1) (1985), 69–80.CrossRefGoogle Scholar
Zagier, D., ‘Nombres de classes et fractions continues’, inJournées Arithmétiques de Bordeaux (Conference, Univ. Bordeaux, 1974), Astérisque 24–25 (1975), 81–97.Google Scholar