The geometry of frieze patterns

Broline, Duane M; Crowe, Donald W.; Isaacs, I. M.

doi:10.1007/bf00183208

Cited by 25 publications

(43 citation statements)

References 2 publications

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“…There are -1 zero-closure, -1 one-closure: the sink 4, -2 two-closures: (1,2), (4,5), -1 three-closures: (1,2,3), -2 four-closures: (1,2,3,4), (1,2,4,5), -1 five-closure. We assign the following initial values to the sides and diagonals of T rs , see Figure 3.…”

Section: Appendix a Jones Polynomial And Q-continued Fractionsmentioning

confidence: 99%

-Deformed Rationals and -Continued Fractions

Morier-Genoud

Ovsienko

2020

Forum of Mathematics, Sigma

105

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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 identitiy 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.

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Section: Appendix a Jones Polynomial And Q-continued Fractionsmentioning

confidence: 99%

-Deformed Rationals and -Continued Fractions

Morier-Genoud

Ovsienko

2020

Forum of Mathematics, Sigma

105

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“…(2) Consider the following frieze with coefficients on a square: This square can not be cut out of a classic Conway-Coxeter frieze because it contains a triangle with values 1, 2, 2, which can not come from a classic Conway-Coxeter frieze by part (1).…”

Section: Frieze Patterns From Subpolygonsmentioning

confidence: 99%

“…We claim that the classic Conway-Coxeter frieze to this triangulation of the m-gon has the desired properties. For this, we use a well-known combinatorial algorithm for computing arbitrary frieze entries from the triangulation, see [1], starting from the vertex 1. The values attached to the vertices by this algorithm are then the frieze entries c 1,j .…”

Section: Frieze Patterns From Subpolygonsmentioning

confidence: 99%

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Frieze Patterns With Coefficients

Cuntz

Holm

Jørgensen

2020

Forum of Mathematics, Sigma

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Frieze patterns, as introduced by Coxeter in the 1970's, are closely related to cluster algebras without coefficients. A suitable generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in an unpublished manuscript by Propp. In this paper we study these frieze patterns with coefficients systematically and prove various fundamental results, generalizing classic results for frieze patterns. As a consequence we see how frieze patterns with coefficients can be obtained from classic frieze patterns by cutting out subpolygons from the triangulated polygons associated to classic Conway-Coxeter frieze patterns. We address the question of which frieze patterns with coefficients can be obtained in this way and solve this problem completely for triangles. Finally, we prove a finiteness result for frieze patterns with coefficients by showing that for a given boundary sequence there are only finitely many (non-zero) frieze patterns with coefficients with entries in a discrete subset of the complex numbers.2010 Mathematics Subject Classification. 05E15, 05E99, 13F60, 51M20.

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“…If T is the frieze table, with entires (i,j), obtained by rule (1) when t is the second row, then T is called a frieze pattern. Many special properties of frieze patterns are derived in [1]- [3]. In particular, we will need the facts (3) (n,j) =0, 1 ~<j~< n, and (4) (n -j,j) = (j, n), 1 ~<j ~< n -1.…”

Section: The Explanationmentioning

confidence: 99%