Infinite friezes

Baur, Karin; Parsons, Mark James; Tschabold, Manuela

doi:10.1016/j.ejc.2015.12.015

Cited by 23 publications

(51 citation statements)

References 16 publications

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“…6 of[BPT16] for a description on how to draw a triangulation of a once-punctured disk as an asymptotic triangulation in the infinite strip. See an example inFig.…”

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confidence: 99%

Cluster algebraic interpretation of infinite friezes

Gunawan

Musiker

Vogel

2019

European Journal of Combinatorics

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Originally studied by Conway and Coxeter, friezes appeared in various recreational mathematics publications in the 1970s. More recently, in 2015, Baur, Parsons, and Tschabold constructed periodic infinite friezes and related them to matching numbers in the once-punctured disk and annulus. In this paper, we study such infinite friezes with an eye towards cluster algebras of type D and affine A, respectively. By examining infinite friezes with Laurent polynomial entries, we discover new symmetries and formulas relating the entries of this frieze to one another. Lastly, we also present a correspondence between Broline, Crowe and Isaacs's classical matching tuples and combinatorial interpretations of elements of cluster algebras from surfaces.Date: June 1, 2017. 2010 Mathematics Subject Classification. 13F60 (primary), 05C70, 05E15 (secondary).

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“…6 of[BPT16] for a description on how to draw a triangulation of a once-punctured disk as an asymptotic triangulation in the infinite strip. See an example inFig.…”

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confidence: 99%

Cluster algebraic interpretation of infinite friezes

Gunawan

Musiker

Vogel

2019

European Journal of Combinatorics

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“…Theorem 4.1 gives an explicit construction of such a characteristic sequence by gluing quiddity cycles together. These sequences also correspond to certain infinite friezes (see [1]).…”

Section: Introductionmentioning

confidence: 95%

“…1 contains p1, 2, 2, 4q. 1 Quiddity cycles are considered up to the action of the dihedral group, for example p1, 2, 3, 1, 2, 3q " p2, 1, 3, 2, 1, 3q. As an application, we reproduce the classification of [12] 2 : Nichols algebras of diagonal type are straight forward generalizations of certain algebras investigated by Lusztig in the classical theory.…”

Section: Introductionmentioning

confidence: 99%

On subsequences of quiddity cycles and Nichols algebras

Cuntz

2018

Journal of Algebra

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“…SL 2 -tilings were generalised by Bergeron and Reutenauer in [5] to SL k -tilings. Other types of SL 2 -tilings, relaxing parts of the definition, were obtained by Baur, Parsons, and Tschabold in [4], Morier-Genoud, Ovsienko, and Tabachnikov in [22], Tschabold in [24], and also in [18] and [20].…”

Section: Introductionmentioning

confidence: 99%

All SL2-tilings come from infinite triangulations

Bessenrodt

Holm

Jørgensen

2017

Advances in Mathematics

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Abstract. An SL 2 -tiling is a bi-infinite matrix of positive integers such that each adjacent 2 × 2-submatrix has determinant 1. Such tilings are infinite analogues of Conway-Coxeter friezes, and they have strong links to cluster algebras, combinatorics, mathematical physics, and representation theory.We show that, by means of so-called Conway-Coxeter counting, every SL 2 -tiling arises from a triangulation of the disc with two, three or four accumulation points.This improves earlier results which only discovered SL 2 -tilings with infinitely many entries equal to 1. Indeed, our methods show that there are large classes of tilings with only finitely many entries equal to 1, including a class of tilings with no 1's at all. In the latter case, we show that the minimal entry of a tiling is unique. [18] are rather special, because they have infinitely many entries equal to 1. Our methods reveal that there are large classes of SL 2 -tilings with only finitely many 1's, and even a class of tilings with no 1's at all, see Remark 3.10. IntroductionIn the latter case, we show that the minimal entry of a tiling is unique, see Lemma 12.4.Motivations for studying SL 2 -tilings. The introduction of SL 2 -tilings in [3] was motivated by applications to linear recurrence relations for certain friezes, and to formulae for cluster variables in Euclidean type, see [3, secs. 7 and 8]. There is an application by Assem and Reutenauer in [2] to formulae for cluster seeds in types A andÃ.

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Infinite friezes

Cited by 23 publications

References 16 publications

Cluster algebraic interpretation of infinite friezes

Cluster algebraic interpretation of infinite friezes

On subsequences of quiddity cycles and Nichols algebras

All SL2-tilings come from infinite triangulations

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