# Partitions of unity in \(\mathrm {SL}(2,\mathbb Z)\), negative continued fractions, and dissections of polygons

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## Abstract

We characterize sequences of positive integers \((a_1,a_2,\ldots ,a_n)\) for which the \(2\times 2\) matrix \(\left( \begin{array}{ll} a_n&{} \quad -\,1\\ 1&{}\quad 0 \end{array} \right) \left( \begin{array}{ll} a_{n-1}&{}\quad -\,1\\ 1&{}\quad 0 \end{array} \right) \cdots \left( \begin{array}{ll} a_1&{}\quad -\,1\\ 1&{}\quad 0 \end{array} \right) \) is either the identity matrix \(\mathrm {Id}\), its negative \(-\,\mathrm {Id}\), or square root of \(-\,\mathrm {Id}\). This extends a theorem of Conway and Coxeter that classifies such solutions subject to a total positivity restriction.

## Notes

## References

- 1.Aigner, M.: Markov’s Theorem and 100 Years of the Uniqueness Conjecture. A Mathematical Journey from Irrational Numbers to Perfect Matchings. Springer, Cham (2013)MATHGoogle Scholar
- 2.Baur, K., Marsh, R.: Frieze patterns for punctured discs. J. Algebraic Combin.
**30**, 349–379 (2009)MathSciNetCrossRefMATHGoogle Scholar - 3.Baur, K., Parsons, M., Tschabold, M.: Infinite friezes. Eur. J. Combin.
**54**, 220–237 (2016)MathSciNetCrossRefMATHGoogle Scholar - 4.Bergeron, F., Reutenauer, C.: \(SL_k\)-tilings of the plane. Ill. J. Math.
**54**, 263–300 (2010)MATHGoogle Scholar - 5.Bessenrodt, C., Holm, T., Jorgensen, P.: Generalized frieze pattern determinants and higher angulations of polygons. J. Combin. Theory Ser. A
**123**, 30–42 (2014)MathSciNetCrossRefMATHGoogle Scholar - 6.Conley, C., Ovsienko, V.: Rotundus: triangulations, Chebyshev polynomials, and Pfaffians. Math. Intell. arXiv:1707.09106
**(to appear)** - 7.Conway, J.H., Coxeter, H.S.M.: Triangulated polygons and frieze patterns. Math. Gaz.
**57**, 87–94 (1973). and 175–183MathSciNetCrossRefMATHGoogle Scholar - 8.Coxeter, H.S.M.: Frieze patterns. Acta Arith.
**18**, 297–310 (1971)MathSciNetCrossRefMATHGoogle Scholar - 9.Coxeter, H.S.M., Rigby, J.F.: Frieze patterns, triangulated polygons and dichromatic symmetry. In: Guy, R.K., Woodrow, E. (eds.) The Lighter Side of Mathematics. MAA Spectrum, pp. 15–27. Mathematical Association of America, Washington, DC (1994)Google Scholar
- 10.Cuntz, M.: On wild Frieze patterns. Exp. Math.
**26**, 342–348 (2017)MathSciNetCrossRefMATHGoogle Scholar - 11.Cuntz, M., Heckenberger, I.: I Weyl groupoids of rank two and continued fractions. Algebra Number Theory
**3**, 317–340 (2009)MathSciNetCrossRefMATHGoogle Scholar - 12.Delyon, F., Souillard, B.: The rotation number for finite difference operators and its properties. Commun. Math. Phys.
**89**, 415–426 (1983)MathSciNetCrossRefMATHGoogle Scholar - 13.Flajolet, P., Noy, M.: Analytic combinatorics of non-crossing configurations. Discrete Math.
**204**, 203–229 (1999)MathSciNetCrossRefMATHGoogle Scholar - 14.Hall, R., Shiu, P.: The index of a Farey sequence. Mich. Math. J.
**51**, 209–223 (2003)MathSciNetCrossRefMATHGoogle Scholar - 15.Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers. Revised by D. R. Heath-Brown and J. H. Silverman. With a foreword by Andrew Wiles, 6th edn, p. 621. Oxford University Press, Oxford (2008)Google Scholar
- 16.Henry, C.-S.: Coxeter friezes and triangulations of polygons. Am. Math. Mon.
**120**, 553–558 (2013)MathSciNetCrossRefMATHGoogle Scholar - 17.Holm, T., Jorgensen, P.: A \(p\)-angulated generalisation of Conway and Coxeter’s theorem on frieze patterns. arXiv:1709.09861
- 18.Krichever, I.: Commuting difference operators and the combinatorial Gale transform. Funct. Anal. Appl.
**49**, 175–188 (2015)MathSciNetCrossRefMATHGoogle Scholar - 19.Morier-Genoud, S.: Coxeter’s frieze patterns at the crossroads of algebra, geometry and combinatorics. Bull. Lond. Math. Soc.
**47**, 895–938 (2015)MathSciNetCrossRefMATHGoogle Scholar - 20.Morier-Genoud, S., Ovsienko, V., Schwartz, R., Tabachnikov, S.: Linear difference equations, frieze patterns and combinatorial Gale transform. Forum Math. Sigma
**2**, e22 (2014)MathSciNetCrossRefMATHGoogle Scholar - 21.Morier-Genoud, S., Ovsienko, V., Tabachnikov, S.: \(2\)-Frieze patterns and the cluster structure of the space of polygons. Ann. Inst. Fourier
**62**, 937–987 (2012)MathSciNetCrossRefMATHGoogle Scholar - 22.Morier-Genoud, S., Ovsienko, V., Tabachnikov, S.: \(\text{ SL }_{2}(\mathbb{R})\)-tilings of the torus, Coxeter–Conway friezes and Farey triangulations. Enseign. Math.
**61**, 71–92 (2015)MathSciNetCrossRefMATHGoogle Scholar - 23.Simon, B.: Sturm Oscillation and Comparison Theorems, Sturm–Liouville theory, pp. 29–43. Birkhäuser, Basel (2005)MATHGoogle Scholar
- 24.Sklyanin, E.K.: The Quantum Toda Chain. Nonlinear Equations in Classical and Quantum Field Theory. Lecture Notes in Physics, vol. 226, pp. 196–233. Springer, Berlin (1985)CrossRefGoogle Scholar
- 25.Zagier, D.: Nombres de classes et fractions continues. In: Journées Arithmétiques de Bordeaux (Conference, Univ. Bordeaux, 1974) Astérisque, vol. 24–25, pp. 81–97 (1975)Google Scholar

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