Friezes
Some (mostly on-line) resources
Here, we collect the links topicwise (each paper listed once only - for one of its topics)
Click here for the same papers in the alphabetic order
Books:
- H.S.M.Coxeter, Regular complex polytopes, Cambridge University Press, Cambridge, 1974.
Introductory papers:
- J.H.Conway, H.S.M.Coxeter, Triangulated polygons and frieze patterns, ---- (Questions about frieze patterns).
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J.H.Conway, H.S.M. Coxeter, Triangulated polygons and frieze patterns (continued), ---- (Solutions).
- H.S.M.Coxeter, Frieze patterns. ----
- D. Broline, D.W. Crowe, I.M.Isaacs, The geometry of frieze patterns. ----
- H.S.M.Coxeter, J.F.Rigby, Frieze patterns, triangulated polygons and dichromatic symmetry . ----
- C.-S. Henry, Coxeter Friezes and Triangulations of Polygons.
- T. Holm, Friezes and tilings.
- M. Pressland, From frieze patterns to cluster categories.
- S. Morier-Genoud, Lecture notes on Integrable Systems and Friezes. ----
- K. Baur, Frieze patterns of integers.
- E. Smirnov, Friezes and continued fractions (in Russian).
Introductory videos:
Survey paper:
Friezes and Farey triangulation:
- S. Morier-Genoud, V. Ovsienko, S. Tabachnikov, SL2(Z)- tilings of the torus, Coxeter-Conway friezes and Farey Triangulations. ----
- V. Ovsienko,
Partitions of unity in SL(2,Z), negative continued fractions, and dissections of polygons. ----
- S. Morier-Genoud, V. Ovsienko,
Farey boat I. Continued fractions and triangulations, modular group and polygon dissections . ----
- I. Short, Classifying SL2-tilings. ----
- A. Felikson, O. Karpenkov, K. Serhiyenko, P. Tumarkin, 3D Farey graph, lambda lengths and SL2-tilings. ----
- I. Short, M. Van Son, A. Zabolotski, Frieze patterns and Farey complexes. ----
- S. Benzaira, I. Short, M. Van Son, A. Zabolotskii, Enumerating tame friezes over ℤ/nℤ. ----
Friezes and cluster algebras, categorification:
- P. Caldero, F. Chapoton, Cluster algebras as Hall algebras of quiver representations. ----
- J. Propp, The combinatorics of frieze patterns and Markoff numbers. ----
- I. Assem, C. Reutenauer, D. Smith, Friezes. ----
- I. Assem, G. Dupont, Friezes and a construction of the euclidean cluster variables Friezes and a construction of the euclidean cluster variables . ----
- I. Assem, G. Dupont, R. Schiffler, D. Smith, Friezes, Strings and Cluster Variables. ----
- L. Guo, On tropical friezes associated with Dynkin diagrams. ----
- K. Baur, B. Marsh, Categorification of a frieze pattern determinant . ----
- V. Ovsienko, S. Tabachnikov, Coxeter's frieze patterns and discretization of the Virasoro orbit. ----
- T. Holm, P. Jorgensen, Generalised friezes and a modified Caldero-Chapoton map depending on a rigid object . ----
- T. Holm, P. Jorgensen, Generalised friezes and a modified Caldero-Chapoton map depending on a rigid object, II . ----
- K. Baur, E. Faber, S. Gratz, K. Serhiyenko, G. Todorov, Mutation of friezes. ----
- K. Baur, E. Faber, S. Gratz, K. Serhiyenko, G. Todorov, Conway-Coxeter friezes and mutation: a survey . ----
- E. Gunawan, G. Muller, Superunitary regions of cluster algebras. ----
Friezes from surfaces:
- K. Baur, B.R. Marsh, Frieze patterns for punctured discs. ----
- M. Tschabold, Arithmetic infinite friezes from punctured discs. ----
- K. Baur, M. Parsons, M. Tschabold, Infinite friezes. ----
- K. Baur, K. Fellner, M. Parsons, M. Tschabold, Growth behaviour of periodic tame friezes. ----
- B. Fontaine, P-G. Plamondon, Counting friezes in type Dn. ----
- E. Gunawan, G. Musiker, H. Vogel, Cluster algebraic interpretation of infinite friezes. ----
- E. Gunawan, R. Schiffler, Frieze vectors and unitary friezes. ----
- K. Baur, I. Canakci, K. Jacobsen, M. Kulkarni, G. Todorov, Infinite friezes and triangulations of annuli . ----
- I. Canakci, A. Felikson, A. Garsia Elsener, P. Tumarkin, Friezes for a pair of pants. ----
- K. Baur, L. Bittmann, E. Gunawan, G. Todorov, E. Yıldırım, Infinite friezes of affine type D . ----
- A. Felikson, P. Tumarkin, Frizes from surfaces and Farey triangulation. ----
Further results on friezes of finite type
Polygon dissections, infinite triangulations:
Weak freizes:
SLk-tilings
- F. Bergeron, C. Reutenauer, SLk-tilings of the plane. ----
- S. Morier-Genoud, V. Ovsienko, S. Tabachnikov, R. Schwartz, Linear difference equations, frieze patterns and combinatorial Gale transform. ----
- M. Cuntz, On wild Frieze patterns. ----
- K. Baur, E. Faber, S. Gratz, K. Serhiyenko, G. Todorov, Friezes satisfying higher SLk-determinants . ----
- Z. Peterson, K. Serhiyenko, SLk-tilings and paths in Zk. ----
Friezes with coefficients:
Non-integer friezes:
- M. Cuntz, A combinatorial model for tame frieze patterns . ----
- T. Holm, P. Jorgensen, A p-angulated generalisation of Conway and Coxeter's theorem on frieze patterns. ----
- M. Cuntz, T. Holm, Frieze patterns over integers and other subsets of the complex numbers . ----
- S. Morier-Genoud Counting Coxeter's friezes over a finite field via moduli spaces . ----
- Amy Tao, Zhichun Joy Zhang, Friezes over Z[√2] and Z[√3]. ----
- M. Cuntz, T. Holm, C. Pagano, Frieze patterns over algebraic numbers. ----
- E. Banaian, L. Farrell, A. Tao, K. Wright, J. Zh. Zhang, Friezes over Z[√2] . ----
- B. Böhmler, M. Cuntz, Frieze patterns over finite commutative local rings . ----
2-friezes, q-deformed friezes, superfriezes:
- J-Ph. Burelle, G. Dupont, Quantum frieze patterns in quantum cluster algebras of type A . ----
- S. Morier-Genoud, V. Ovsienko, S. Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons. ----
- S. Morier-Genoud, Arithmetics of 2-friezes. ----
- S. Morier-Genoud, V. Ovsienko, S. Tabachnikov, Introducing supersymmetric frieze patterns and linear difference operators. ----
- S. Morier-Genoud, V. Ovsienko, Quantum real numbers and q-deformed Conway-Coxeter friezes . ----
Symplectic and Non-commutative friezes:
Y-friezes, additive friezes:
Pentagramma Mirificum:
Introduction to cluster algebras:
Some slides:
Software: