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On surface cluster algebras Ilke C anak c Surface cluster algebras On surface cluster algebras: Snake graph Abstract Snake Graphs Relation to calculus and dreaded torus Cluster Algebras Self-crossing snake graphs Application

  1. On surface cluster algebras Overview ˙ Ilke C ¸anak¸ cı • Cluster algebras from surfaces , introduced in [FST], have a Surface cluster algebras geometric interpretation in surfaces. Abstract Snake Graphs • A surface cluster algebra A is associated to a surface S with Relation to boundary that has finitely many marked points. Cluster Algebras Self-crossing snake graphs 2 Application 1 3 • Cluster variables are in bijection with certain curves [FST] , called arcs . Two crossing arcs satisfy the skein relations , [MW] . • The authors in [MSW] associate a connected graph, called the snake graph to each arc in the surface to obtain a direct formula, the expansion formula , for cluster variables of surface cluster algebras. 1 � x γ = x ( P ) y ( P ) cross ( γ, T ) P ⊢ G γ ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 4 / 35

  2. On surface cluster algebras Overview ˙ Ilke C ¸anak¸ cı • Cluster algebras from surfaces , introduced in [FST], have a Surface cluster algebras geometric interpretation in surfaces. Abstract Snake Graphs • A surface cluster algebra A is associated to a surface S with Relation to boundary that has finitely many marked points. Cluster Algebras Self-crossing snake graphs 2 Application 1 3 • Cluster variables are in bijection with certain curves [FST] , called arcs . Two crossing arcs satisfy the skein relations , [MW] . • The authors in [MSW] associate a connected graph, called the snake graph to each arc in the surface to obtain a direct formula, the expansion formula , for cluster variables of surface cluster algebras. 1 � x γ = x ( P ) y ( P ) cross ( γ, T ) P ⊢ G γ ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 4 / 35

  3. On surface cluster algebras Overview ˙ Ilke C ¸anak¸ cı • Cluster algebras from surfaces , introduced in [FST], have a Surface cluster algebras geometric interpretation in surfaces. Abstract Snake Graphs • A surface cluster algebra A is associated to a surface S with Relation to boundary that has finitely many marked points. Cluster Algebras Self-crossing snake graphs 2 Application γ 1 1 3 • Cluster variables are in bijection with certain curves [FST] , called arcs . Two crossing arcs satisfy the skein relations , [MW] . • The authors in [MSW] associate a connected graph, called the snake graph to each arc in the surface to obtain a direct formula, the expansion formula , for cluster variables of surface cluster algebras. 1 � x γ = x ( P ) y ( P ) cross ( γ, T ) P ⊢ G γ ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 4 / 35

  4. On surface cluster algebras Overview ˙ Ilke C ¸anak¸ cı • Cluster algebras from surfaces , introduced in [FST], have a Surface cluster algebras geometric interpretation in surfaces. Abstract Snake Graphs • A surface cluster algebra A is associated to a surface S with Relation to boundary that has finitely many marked points. Cluster Algebras Self-crossing snake graphs 2 Application γ 2 γ 1 1 3 • Cluster variables are in bijection with certain curves [FST] , called arcs . Two crossing arcs satisfy the skein relations , [MW] . • The authors in [MSW] associate a connected graph, called the snake graph to each arc in the surface to obtain a direct formula, the expansion formula , for cluster variables of surface cluster algebras. 1 � x γ = x ( P ) y ( P ) cross ( γ, T ) P ⊢ G γ ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 4 / 35

  5. On surface cluster algebras Overview ˙ Ilke C ¸anak¸ cı • Cluster algebras from surfaces , introduced in [FST], have a Surface cluster algebras geometric interpretation in surfaces. Abstract Snake Graphs • A surface cluster algebra A is associated to a surface S with Relation to boundary that has finitely many marked points. Cluster Algebras Self-crossing snake graphs 2 Application γ 2 γ 1 1 3 • Cluster variables are in bijection with certain curves [FST] , called arcs . Two crossing arcs satisfy the skein relations , [MW] . • The authors in [MSW] associate a connected graph, called the snake graph to each arc in the surface to obtain a direct formula, the expansion formula , for cluster variables of surface cluster algebras. 1 � x γ = x ( P ) y ( P ) cross ( γ, T ) P ⊢ G γ ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 4 / 35

  6. On surface cluster algebras Overview ˙ Ilke C ¸anak¸ cı • Cluster algebras from surfaces , introduced in [FST], have a Surface cluster algebras geometric interpretation in surfaces. Abstract Snake Graphs • A surface cluster algebra A is associated to a surface S with Relation to boundary that has finitely many marked points. Cluster Algebras Self-crossing snake graphs γ 5 γ 4 2 Application γ 2 γ 1 1 γ 3 3 γ 6 • Cluster variables are in bijection with certain curves [FST] , called arcs . Two crossing arcs satisfy the skein relations , [MW] . • The authors in [MSW] associate a connected graph, called the snake graph to each arc in the surface to obtain a direct formula, the expansion formula , for cluster variables of surface cluster algebras. 1 � x γ = x ( P ) y ( P ) cross ( γ, T ) P ⊢ G γ ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 4 / 35

  7. On surface cluster algebras Overview ˙ Ilke C ¸anak¸ cı • Cluster algebras from surfaces , introduced in [FST], have a Surface cluster algebras geometric interpretation in surfaces. Abstract Snake Graphs • A surface cluster algebra A is associated to a surface S with Relation to boundary that has finitely many marked points. Cluster Algebras Self-crossing snake graphs γ 5 γ 4 2 Application γ 2 x γ 1 x γ 2 = ∗ x γ 3 x γ 4 + ∗ x γ 5 x γ 6 γ 1 1 γ 3 3 γ 6 Skein relation ( [MW]) • Cluster variables are in bijection with certain curves [FST] , called arcs . Two crossing arcs satisfy the skein relations , [MW] . • The authors in [MSW] associate a connected graph, called the snake graph to each arc in the surface to obtain a direct formula, the expansion formula , for cluster variables of surface cluster algebras. 1 � x γ = x ( P ) y ( P ) cross ( γ, T ) P ⊢ G γ ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 4 / 35

  8. On surface cluster algebras Overview ˙ Ilke C ¸anak¸ cı • Cluster algebras from surfaces , introduced in [FST], have a Surface cluster algebras geometric interpretation in surfaces. Abstract Snake Graphs • A surface cluster algebra A is associated to a surface S with Relation to boundary that has finitely many marked points. Cluster Algebras Self-crossing snake graphs γ 5 γ 4 2 Application γ 2 x γ 1 x γ 2 = ∗ x γ 3 x γ 4 + ∗ x γ 5 x γ 6 γ 1 1 γ 3 3 γ 6 Skein relation ( [MW]) • Cluster variables are in bijection with certain curves [FST] , called arcs . Two crossing arcs satisfy the skein relations , [MW] . • The authors in [MSW] associate a connected graph, called the snake graph to each arc in the surface to obtain a direct formula, the expansion formula , for cluster variables of surface cluster algebras. 1 � x γ = x ( P ) y ( P ) cross ( γ, T ) P ⊢ G γ ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 4 / 35

  9. On surface cluster algebras Motivation ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Graphs Let A ( S , M ) cluster algebra associated to a surface ( S , M ) . Relation to Cluster Algebras Self-crossing We have the following situation: snake graphs Application Question “How much can we recover from snake graphs themselves?” In particular, • When do the two arcs corresponding to two snake graphs cross? • What are the snake graphs corresponding to the skein relations? ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 5 / 35

  10. On surface cluster algebras Motivation ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Graphs Let A ( S , M ) cluster algebra associated to a surface ( S , M ) . Relation to Cluster Algebras Self-crossing We have the following situation: snake graphs Application [FST] ← → arc cluster variable Question “How much can we recover from snake graphs themselves?” In particular, • When do the two arcs corresponding to two snake graphs cross? • What are the snake graphs corresponding to the skein relations? ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 5 / 35

  11. On surface cluster algebras Motivation ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Graphs Let A ( S , M ) cluster algebra associated to a surface ( S , M ) . Relation to Cluster Algebras We have the following situation: Self-crossing snake graphs Application [FST] [MSW] ← → arc − → snake graph cluster variable Question “How much can we recover from snake graphs themselves?” In particular, • When do the two arcs corresponding to two snake graphs cross? • What are the snake graphs corresponding to the skein relations? ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 5 / 35

  12. On surface cluster algebras Motivation ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Graphs Let A ( S , M ) cluster algebra associated to a surface ( S , M ) . Relation to Cluster Algebras We have the following situation: Self-crossing snake graphs Application [FST] [MSW] ← → arc − → snake graph cluster variable Question “How much can we recover from snake graphs themselves?” In particular, • When do the two arcs corresponding to two snake graphs cross? • What are the snake graphs corresponding to the skein relations? ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 5 / 35

  13. On surface cluster algebras Motivation ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Graphs Let A ( S , M ) cluster algebra associated to a surface ( S , M ) . Relation to Cluster Algebras We have the following situation: Self-crossing snake graphs Application [FST] [MSW] ← → arc − → snake graph cluster variable Question “How much can we recover from snake graphs themselves?” In particular, • When do the two arcs corresponding to two snake graphs cross? • What are the snake graphs corresponding to the skein relations? ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 5 / 35

  14. On surface cluster algebras Motivation ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Graphs Let A ( S , M ) cluster algebra associated to a surface ( S , M ) . Relation to Cluster Algebras We have the following situation: Self-crossing snake graphs Application [FST] [MSW] ← → arc − → snake graph cluster variable Question “How much can we recover from snake graphs themselves?” In particular, • When do the two arcs corresponding to two snake graphs cross? • What are the snake graphs corresponding to the skein relations? ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 5 / 35

  15. On surface cluster algebras Motivation ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Graphs Let A ( S , M ) cluster algebra associated to a surface ( S , M ) . Relation to Cluster Algebras We have the following situation: Self-crossing snake graphs Application [FST] [MSW] ← → arc − → snake graph cluster variable Question “How much can we recover from snake graphs themselves?” In particular, • When do the two arcs corresponding to two snake graphs cross? • What are the snake graphs corresponding to the skein relations? ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 5 / 35

  16. On surface cluster algebras Surface Cluster Algebras ˙ Ilke C ¸anak¸ cı Surface cluster • Let S be a connected oriented 2-dimensional Riemann surface algebras with nonempty boundary, and let M be a nonempty finite subset Abstract Snake Graphs of the boundary of S , such that each boundary component of S Relation to Cluster Algebras contains at least one point of M . The elements of M are called Self-crossing marked points . The pair ( S , M ) is called a bordered surface snake graphs with marked points . Application b = 1 b = 2 g = 0 g = 1 g = 2 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 6 / 35

  17. On surface cluster algebras Surface Cluster Algebras ˙ Ilke C ¸anak¸ cı Surface cluster • Let S be a connected oriented 2-dimensional Riemann surface algebras with nonempty boundary, and let M be a nonempty finite subset Abstract Snake Graphs of the boundary of S , such that each boundary component of S Relation to Cluster Algebras contains at least one point of M . The elements of M are called Self-crossing marked points . The pair ( S , M ) is called a bordered surface snake graphs with marked points . Application b = 1 b = 2 g = 0 g = 1 g = 2 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 6 / 35

  18. On surface cluster algebras Surface Cluster Algebras ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Definition Graphs An arc γ in ( S , M ) is a curve in S , considered up to isotopy, such Relation to Cluster Algebras that: Self-crossing snake graphs • the endpoints of γ are in M ; Application • γ does not cross itself; • except for the endpoints, γ is disjoint from the boundary of S ; and • γ does not cut out a monogon or a bigon. Remark Curves that connect two marked points and lie entirely on the boundary of S without passing through a third marked point are boundary segments . Note that boundary segments are not arcs. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 7 / 35

  19. On surface cluster algebras Surface Cluster Algebras ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Definition Graphs An arc γ in ( S , M ) is a curve in S , considered up to isotopy, such Relation to Cluster Algebras that: Self-crossing snake graphs • the endpoints of γ are in M ; Application • γ does not cross itself; • except for the endpoints, γ is disjoint from the boundary of S ; and • γ does not cut out a monogon or a bigon. Remark Curves that connect two marked points and lie entirely on the boundary of S without passing through a third marked point are boundary segments . Note that boundary segments are not arcs. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 7 / 35

  20. On surface cluster algebras Surface Cluster Algebras ˙ Ilke C ¸anak¸ cı Surface cluster algebras Definition Abstract Snake For any two arcs γ, γ ′ in S , let e ( γ, γ ′ ) be the minimal number of Graphs crossings of arcs α and α ′ , where α and α ′ range over all arcs Relation to Cluster Algebras isotopic to γ and γ ′ , respectively. We say that arcs γ and γ ′ are Self-crossing snake graphs compatible if e ( γ, γ ′ ) = 0. Application Definition A triangulation is a maximal collection of pairwise compatible arcs (together with all boundary segments). 1 2 3 4 5 6 7 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 8 / 35

  21. On surface cluster algebras Surface Cluster Algebras ˙ Ilke C ¸anak¸ cı Surface cluster algebras Definition Abstract Snake For any two arcs γ, γ ′ in S , let e ( γ, γ ′ ) be the minimal number of Graphs crossings of arcs α and α ′ , where α and α ′ range over all arcs Relation to Cluster Algebras isotopic to γ and γ ′ , respectively. We say that arcs γ and γ ′ are Self-crossing snake graphs compatible if e ( γ, γ ′ ) = 0. Application Definition A triangulation is a maximal collection of pairwise compatible arcs (together with all boundary segments). 1 2 3 4 5 6 7 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 8 / 35

  22. On surface cluster algebras Surface Cluster Algebras ˙ Ilke C ¸anak¸ cı Surface cluster algebras Definition Abstract Snake For any two arcs γ, γ ′ in S , let e ( γ, γ ′ ) be the minimal number of Graphs crossings of arcs α and α ′ , where α and α ′ range over all arcs Relation to Cluster Algebras isotopic to γ and γ ′ , respectively. We say that arcs γ and γ ′ are Self-crossing snake graphs compatible if e ( γ, γ ′ ) = 0. Application Definition A triangulation is a maximal collection of pairwise compatible arcs (together with all boundary segments). 1 2 3 4 5 6 7 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 8 / 35

  23. On surface cluster algebras Surface Cluster Algebras ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Graphs Relation to Cluster Algebras Self-crossing Definition snake graphs Triangulations are connected to each other by sequences of flips . Application Each flip replaces a single arc γ in a triangulation T by a (unique) arc γ ′ � = γ that, together with the remaining arcs in T , forms a new triangulation. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 9 / 35

  24. On surface cluster algebras Surface Cluster Algebras ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Graphs Definition Relation to Cluster Algebras Triangulations are connected to each other by sequences of flips . Self-crossing Each flip replaces a single arc γ in a triangulation T by a (unique) snake graphs arc γ ′ � = γ that, together with the remaining arcs in T , forms a new Application triangulation. a d τ k b c ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 9 / 35

  25. On surface cluster algebras Surface Cluster Algebras ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Graphs Definition Relation to Cluster Algebras Triangulations are connected to each other by sequences of flips . Self-crossing Each flip replaces a single arc γ in a triangulation T by a (unique) snake graphs arc γ ′ � = γ that, together with the remaining arcs in T , forms a new Application triangulation. a a d d τ ′ k τ k b c b c ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 9 / 35

  26. On surface cluster algebras Surface Cluster Algebras ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Graphs Definition Relation to Cluster Algebras Triangulations are connected to each other by sequences of flips . Self-crossing Each flip replaces a single arc γ in a triangulation T by a (unique) snake graphs arc γ ′ � = γ that, together with the remaining arcs in T , forms a new Application triangulation. 1 2 3 4 5 6 7 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 9 / 35

  27. On surface cluster algebras Surface Cluster Algebras ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Graphs Definition Relation to Cluster Algebras Triangulations are connected to each other by sequences of flips . Self-crossing Each flip replaces a single arc γ in a triangulation T by a (unique) snake graphs arc γ ′ � = γ that, together with the remaining arcs in T , forms a new Application triangulation. 1 2 3 3 4 5 6 7 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 9 / 35

  28. On surface cluster algebras Surface Cluster Algebras ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Graphs Definition Relation to Cluster Algebras Triangulations are connected to each other by sequences of flips . Self-crossing Each flip replaces a single arc γ in a triangulation T by a (unique) snake graphs arc γ ′ � = γ that, together with the remaining arcs in T , forms a new Application triangulation. 1 1 2 2 3 3 4 4 5 5 6 6 7 7 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 9 / 35

  29. On surface cluster algebras Surface Cluster Algebras ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Graphs Definition Relation to Cluster Algebras Triangulations are connected to each other by sequences of flips . Self-crossing Each flip replaces a single arc γ in a triangulation T by a (unique) snake graphs arc γ ′ � = γ that, together with the remaining arcs in T , forms a new Application triangulation. 1 1 2 2 3 3 4 4 5 5 5 6 6 7 7 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 9 / 35

  30. On surface cluster algebras Surface Cluster Algebras ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Graphs Definition Relation to Triangulations are connected to each other by sequences of flips . Cluster Algebras Each flip replaces a single arc γ in a triangulation T by a (unique) Self-crossing snake graphs arc γ ′ � = γ that, together with the remaining arcs in T , forms a new Application triangulation. 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 9 / 35

  31. On surface cluster algebras Surface Cluster Algebras ˙ Ilke C ¸anak¸ cı Theorem (FST,FT) Surface cluster algebras For cluster algebras from surfaces Abstract Snake Graphs • there are bijections Relation to Cluster Algebras Self-crossing { arcs } − → { cluster variables } snake graphs �→ x γ Application γ { triangulations } − → { clusters } T = { τ 1 , · · · , τ n } �→ x T = { x τ 1 , · · · , x τ n } • The triangulation T \{ τ k } ∪ { τ ′ k } obtained by flipping the arc τ k corresponds to the mutation µ k ( x T ) = x T \{ x τ k } ∪ { x τ ′ k } . Definition The surface cluster algebra A = A ( S , M ) associated to a surface ( S , M ) is a Z -subalgebra of Q ( x 1 , · · · , x n ) generated by all cluster variables x γ . ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 10 / 35

  32. On surface cluster algebras Snake graphs and perfect ˙ Ilke C ¸anak¸ cı matchings Surface cluster algebras Abstract Snake Graphs For each arc γ in a surface ( S , M , T ) , we associate a weighted graph Relation to G γ , called snake graph , from γ and T . Cluster Algebras Self-crossing snake graphs Application 1 2 3 γ 4 5 6 7 A perfect matching P of a graph G is a subset of the set of edges of G such that each vertex of G is incident to exactly one edge in P . ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 11 / 35

  33. On surface cluster algebras Snake graphs and perfect ˙ Ilke C ¸anak¸ cı matchings Surface cluster algebras Abstract Snake Graphs For each arc γ in a surface ( S , M , T ) , we associate a weighted graph Relation to G γ , called snake graph , from γ and T . Cluster Algebras Self-crossing snake graphs Application 1 2 3 γ 4 5 6 7 A perfect matching P of a graph G is a subset of the set of edges of G such that each vertex of G is incident to exactly one edge in P . ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 11 / 35

  34. On surface cluster algebras Snake graphs and perfect ˙ Ilke C ¸anak¸ cı matchings Surface cluster algebras Abstract Snake Graphs For each arc γ in a surface ( S , M , T ) , we associate a weighted graph Relation to G γ , called snake graph , from γ and T . Cluster Algebras Self-crossing snake graphs Application 1 1 2 2 3 γ 4 5 6 7 A perfect matching P of a graph G is a subset of the set of edges of G such that each vertex of G is incident to exactly one edge in P . ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 11 / 35

  35. On surface cluster algebras Snake graphs and perfect ˙ Ilke C ¸anak¸ cı matchings Surface cluster algebras Abstract Snake Graphs For each arc γ in a surface ( S , M , T ) , we associate a weighted graph Relation to G γ , called snake graph , from γ and T . Cluster Algebras Self-crossing snake graphs Application 1 1 2 2 3 2 γ 4 1 5 6 7 A perfect matching P of a graph G is a subset of the set of edges of G such that each vertex of G is incident to exactly one edge in P . ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 11 / 35

  36. On surface cluster algebras Snake graphs and perfect ˙ Ilke C ¸anak¸ cı matchings Surface cluster algebras Abstract Snake Graphs For each arc γ in a surface ( S , M , T ) , we associate a weighted graph Relation to G γ , called snake graph , from γ and T . Cluster Algebras Self-crossing snake graphs Application 1 1 2 2 3 3 2 γ 4 1 2 3 5 1 6 7 A perfect matching P of a graph G is a subset of the set of edges of G such that each vertex of G is incident to exactly one edge in P . ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 11 / 35

  37. On surface cluster algebras Snake graphs and perfect ˙ Ilke C ¸anak¸ cı matchings Surface cluster algebras Abstract Snake Graphs For each arc γ in a surface ( S , M , T ) , we associate a weighted graph Relation to G γ , called snake graph , from γ and T . Cluster Algebras Self-crossing snake graphs Application 1 2 3 2 2 γ 4 1 1 2 2 3 3 5 1 1 6 7 A perfect matching P of a graph G is a subset of the set of edges of G such that each vertex of G is incident to exactly one edge in P . ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 11 / 35

  38. On surface cluster algebras Snake graphs and perfect ˙ Ilke C ¸anak¸ cı matchings Surface cluster algebras Abstract Snake Graphs For each arc γ in a surface ( S , M , T ) , we associate a weighted graph Relation to G γ , called snake graph , from γ and T . Cluster Algebras Self-crossing snake graphs Application 1 2 7 6 4 5 6 3 3 4 5 6 G γ 2 2 7 γ 4 3 4 5 1 2 3 5 1 6 7 A perfect matching P of a graph G is a subset of the set of edges of G such that each vertex of G is incident to exactly one edge in P . ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 11 / 35

  39. On surface cluster algebras Snake graphs and perfect ˙ Ilke C ¸anak¸ cı matchings Surface cluster algebras Abstract Snake Graphs For each arc γ in a surface ( S , M , T ) , we associate a weighted graph Relation to G γ , called snake graph , from γ and T . Cluster Algebras Self-crossing snake graphs Application 1 2 7 6 4 5 6 3 3 4 5 6 G γ 2 2 7 γ 4 3 4 5 1 2 3 5 1 6 7 A perfect matching P of a graph G is a subset of the set of edges of G such that each vertex of G is incident to exactly one edge in P . ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 11 / 35

  40. On surface cluster algebras Snake graphs and perfect ˙ Ilke C ¸anak¸ cı matchings Surface cluster algebras For each arc γ in a surface ( S , M , T ) , we associate a weighted graph Abstract Snake Graphs G γ , called snake graph , from γ and T . Relation to Cluster Algebras Self-crossing snake graphs 1 2 7 6 Application 4 5 6 3 3 4 5 6 G γ 2 2 7 γ 4 3 4 5 1 2 3 5 1 6 7 A perfect matching P of a graph G is a subset of the set of edges of G such that each vertex of G is incident to exactly one edge in P . 7 4 3 4 5 6 7 1 2 3 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 11 / 35

  41. On surface cluster algebras Snake graphs and perfect ˙ Ilke C ¸anak¸ cı matchings Surface cluster algebras For each arc γ in a surface ( S , M , T ) , we associate a weighted graph Abstract Snake Graphs G γ , called snake graph , from γ and T . Relation to Cluster Algebras Self-crossing snake graphs 1 2 7 6 Application 4 5 6 3 3 4 5 6 G γ 2 2 7 γ 4 3 4 5 1 2 3 5 1 6 7 A perfect matching P of a graph G is a subset of the set of edges of G such that each vertex of G is incident to exactly one edge in P . 7 7 4 4 6 3 4 5 6 3 4 5 6 7 7 2 4 1 2 1 2 3 3 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 11 / 35

  42. On surface cluster algebras Expansion formula ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Graphs The authors in [MSW] gives an explicit formula, called expansion Relation to Cluster Algebras formula , for cluster variables. The formula is given by Self-crossing snake graphs 1 Application � x γ = x ( P ) y ( P ) cross ( γ, T ) P ⊢ G γ where the sum is over all perfect matchings P of G γ . 7 7 4 4 6 3 4 5 6 3 4 5 6 7 7 2 4 1 2 1 2 3 3 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 12 / 35

  43. On surface cluster algebras Expansion formula ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Graphs The authors in [MSW] gives an explicit formula, called expansion Relation to Cluster Algebras formula , for cluster variables. The formula is given by Self-crossing snake graphs 1 Application � x γ = x ( P ) y ( P ) cross ( γ, T ) P ⊢ G γ where the sum is over all perfect matchings P of G γ . 7 7 4 4 6 3 4 5 6 3 4 5 6 7 7 2 4 1 2 1 2 3 3 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 12 / 35

  44. On surface cluster algebras Expansion formula ˙ Ilke C ¸anak¸ cı Surface cluster algebras The authors in [MSW] gives an explicit formula, called expansion Abstract Snake Graphs formula , for cluster variables. The formula is given by Relation to Cluster Algebras 1 � Self-crossing x γ = x ( P ) y ( P ) snake graphs cross ( γ, T ) P ⊢ G γ Application where the sum is over all perfect matchings P of G γ . 7 7 4 4 6 3 4 5 6 3 4 5 6 7 7 2 4 1 2 1 2 3 3 x ( P ) = x 3 x 4 x 7 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 12 / 35

  45. On surface cluster algebras Expansion formula ˙ Ilke C ¸anak¸ cı Surface cluster algebras The authors in [MSW] gives an explicit formula, called expansion Abstract Snake Graphs formula , for cluster variables. The formula is given by Relation to Cluster Algebras 1 � Self-crossing x γ = x ( P ) y ( P ) snake graphs cross ( γ, T ) P ⊢ G γ Application where the sum is over all perfect matchings P of G γ . 7 7 4 4 6 3 4 5 6 7 3 4 5 6 7 2 4 1 2 1 2 3 3 x ( P ) = x 2 x 3 x 2 x ( P ) = x 3 x 4 x 7 4 x 6 x 7 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 12 / 35

  46. On surface cluster algebras ˙ Ilke C ¸anak¸ cı Surface cluster 5 1 algebras Abstract Snake Graphs 6 1 5 2 Relation to Cluster Algebras Self-crossing snake graphs 7 1 6 2 5 3 Application 1 7 2 6 3 5 2 7 3 6 4 3 7 4 6 4 7 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 13 / 35

  47. On surface cluster algebras ˙ Ilke C ¸anak¸ cı Surface cluster Applying the formula, the cluster variable corresponding to the arc γ algebras is given by Abstract Snake Graphs Relation to Cluster Algebras 1 ( x 1 x 2 x 3 x 2 5 x 6 + y 4 x 1 x 2 x 5 x 6 + y 7 x 1 x 2 x 3 x 2 = 5 + x γ Self-crossing snake graphs x 1 x 2 x 3 x 4 x 5 x 6 x 7 Application y 3 y 4 x 1 x 4 x 5 x 6 + y 4 y 7 x 1 x 2 x 5 + y 6 y 7 x 1 x 2 x 3 x 5 x 7 + y 2 y 3 y 4 x 3 x 4 x 5 x 6 + y 3 y 4 y 7 x 1 x 4 x 5 + y 4 y 6 y 7 x 1 x 2 x 7 + y 1 y 2 y 3 y 4 x 2 x 3 x 4 x 5 x 6 + y 2 y 3 y 4 y 7 x 3 x 4 x 5 + y 3 y 4 y 6 y 7 x 1 x 4 x 7 + y 4 y 5 y 6 y 7 x 1 x 2 x 4 x 6 x 7 + y 1 y 2 y 3 y 4 y 7 x 2 x 3 x 4 x 5 + y 2 y 3 y 4 y 6 y 7 x 3 x 4 x 7 + y 3 y 4 y 5 y 6 y 7 x 1 x 2 4 x 6 x 7 + y 1 y 2 y 3 y 4 y 6 y 7 x 2 x 3 x 4 x 7 + y 2 y 3 y 4 y 5 y 6 y 7 x 3 x 2 4 x 6 x 7 + y 1 y 2 y 3 y 4 y 5 y 6 y 7 x 2 x 3 x 2 4 x 6 x 7 ) . ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 14 / 35

  48. On surface cluster algebras Our results ˙ Ilke C ¸anak¸ cı Surface cluster algebras • We introduce the notion of an abstract snake graph , which is Abstract Snake Graphs not necessarily related to an arc in a surface. Relation to Cluster Algebras • We define what it means for two abstract snake graphs to Self-crossing cross. snake graphs Application • Given two crossing snake graphs, we construct the resolution of the crossing as two pairs of snake graphs from the original pair of crossing snake graphs. • We then prove that there is a bijection ϕ between the set of perfect matchings of the two crossing snake graphs and the set of perfect matchings of the resolution. • We then apply our constructions to snake graphs arising from unpunctured surfaces. • We then extend our results to self-crossing snake graphs associated to self-crossing arcs in a surface. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 15 / 35

  49. On surface cluster algebras Our results ˙ Ilke C ¸anak¸ cı Surface cluster algebras • We introduce the notion of an abstract snake graph , which is Abstract Snake Graphs not necessarily related to an arc in a surface. Relation to Cluster Algebras • We define what it means for two abstract snake graphs to Self-crossing cross. snake graphs Application • Given two crossing snake graphs, we construct the resolution of the crossing as two pairs of snake graphs from the original pair of crossing snake graphs. • We then prove that there is a bijection ϕ between the set of perfect matchings of the two crossing snake graphs and the set of perfect matchings of the resolution. • We then apply our constructions to snake graphs arising from unpunctured surfaces. • We then extend our results to self-crossing snake graphs associated to self-crossing arcs in a surface. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 15 / 35

  50. On surface cluster algebras Our results ˙ Ilke C ¸anak¸ cı Surface cluster algebras • We introduce the notion of an abstract snake graph , which is Abstract Snake Graphs not necessarily related to an arc in a surface. Relation to Cluster Algebras • We define what it means for two abstract snake graphs to Self-crossing cross. snake graphs Application • Given two crossing snake graphs, we construct the resolution of the crossing as two pairs of snake graphs from the original pair of crossing snake graphs. • We then prove that there is a bijection ϕ between the set of perfect matchings of the two crossing snake graphs and the set of perfect matchings of the resolution. • We then apply our constructions to snake graphs arising from unpunctured surfaces. • We then extend our results to self-crossing snake graphs associated to self-crossing arcs in a surface. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 15 / 35

  51. On surface cluster algebras Our results ˙ Ilke C ¸anak¸ cı Surface cluster algebras • We introduce the notion of an abstract snake graph , which is Abstract Snake Graphs not necessarily related to an arc in a surface. Relation to Cluster Algebras • We define what it means for two abstract snake graphs to Self-crossing cross. snake graphs Application • Given two crossing snake graphs, we construct the resolution of the crossing as two pairs of snake graphs from the original pair of crossing snake graphs. • We then prove that there is a bijection ϕ between the set of perfect matchings of the two crossing snake graphs and the set of perfect matchings of the resolution. • We then apply our constructions to snake graphs arising from unpunctured surfaces. • We then extend our results to self-crossing snake graphs associated to self-crossing arcs in a surface. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 15 / 35

  52. On surface cluster algebras Our results ˙ Ilke C ¸anak¸ cı Surface cluster algebras • We introduce the notion of an abstract snake graph , which is Abstract Snake Graphs not necessarily related to an arc in a surface. Relation to Cluster Algebras • We define what it means for two abstract snake graphs to Self-crossing cross. snake graphs Application • Given two crossing snake graphs, we construct the resolution of the crossing as two pairs of snake graphs from the original pair of crossing snake graphs. • We then prove that there is a bijection ϕ between the set of perfect matchings of the two crossing snake graphs and the set of perfect matchings of the resolution. • We then apply our constructions to snake graphs arising from unpunctured surfaces. • We then extend our results to self-crossing snake graphs associated to self-crossing arcs in a surface. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 15 / 35

  53. On surface cluster algebras Our results ˙ Ilke C ¸anak¸ cı Surface cluster algebras • We introduce the notion of an abstract snake graph , which is Abstract Snake Graphs not necessarily related to an arc in a surface. Relation to Cluster Algebras • We define what it means for two abstract snake graphs to Self-crossing cross. snake graphs Application • Given two crossing snake graphs, we construct the resolution of the crossing as two pairs of snake graphs from the original pair of crossing snake graphs. • We then prove that there is a bijection ϕ between the set of perfect matchings of the two crossing snake graphs and the set of perfect matchings of the resolution. • We then apply our constructions to snake graphs arising from unpunctured surfaces. • We then extend our results to self-crossing snake graphs associated to self-crossing arcs in a surface. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 15 / 35

  54. On surface cluster algebras Our results ˙ Ilke C ¸anak¸ cı Surface cluster algebras • We introduce the notion of an abstract snake graph , which is Abstract Snake Graphs not necessarily related to an arc in a surface. Relation to Cluster Algebras • We define what it means for two abstract snake graphs to Self-crossing cross. snake graphs Application • Given two crossing snake graphs, we construct the resolution of the crossing as two pairs of snake graphs from the original pair of crossing snake graphs. • We then prove that there is a bijection ϕ between the set of perfect matchings of the two crossing snake graphs and the set of perfect matchings of the resolution. • We then apply our constructions to snake graphs arising from unpunctured surfaces. • We then extend our results to self-crossing snake graphs associated to self-crossing arcs in a surface. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 15 / 35

  55. On surface cluster algebras Abstract Snake Graphs ˙ Ilke C ¸anak¸ cı Definition Surface cluster algebras A snake graph G is a connected graph in R 2 consisting of a finite Abstract Snake Graphs sequence of tiles G 1 , G 2 , . . . , G d with d ≥ 1 , such that for each Relation to i = 1 , . . . , d − 1 Cluster Algebras Self-crossing (i) G i and G i +1 share exactly one edge e i and this edge is either the snake graphs north edge of G i and the south edge of G i +1 or the east edge of Application G i and the west edge of G i +1 . (ii) G i and G j have no edge in common whenever | i − j | ≥ 2 . (ii) G i and G j are disjoint whenever | i − j | ≥ 3 . Example G ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 16 / 35

  56. On surface cluster algebras Abstract Snake Graphs ˙ Ilke C ¸anak¸ cı Definition Surface cluster algebras A snake graph G is a connected graph in R 2 consisting of a finite Abstract Snake Graphs sequence of tiles G 1 , G 2 , . . . , G d with d ≥ 1 , such that for each Relation to i = 1 , . . . , d − 1 Cluster Algebras Self-crossing (i) G i and G i +1 share exactly one edge e i and this edge is either the snake graphs north edge of G i and the south edge of G i +1 or the east edge of Application G i and the west edge of G i +1 . (ii) G i and G j have no edge in common whenever | i − j | ≥ 2 . (ii) G i and G j are disjoint whenever | i − j | ≥ 3 . Example G ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 16 / 35

  57. On surface cluster algebras ˙ Ilke C ¸anak¸ cı Example Surface cluster algebras Abstract Snake Graphs Relation to Cluster Algebras Self-crossing snake graphs Application G G 1 Notation • G = ( G 1 , G 2 , . . . , G d ) • G [ i , i + t ] = ( G i , G i +1 , . . . , G i + t ) • We denote by e i the interior edge between the tiles G i and G i +1 . ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 17 / 35

  58. On surface cluster algebras ˙ Ilke C ¸anak¸ cı Example Surface cluster algebras Abstract Snake Graphs Relation to Cluster Algebras Self-crossing snake graphs Application G G 1 G 2 Notation • G = ( G 1 , G 2 , . . . , G d ) • G [ i , i + t ] = ( G i , G i +1 , . . . , G i + t ) • We denote by e i the interior edge between the tiles G i and G i +1 . ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 17 / 35

  59. On surface cluster algebras ˙ Ilke C ¸anak¸ cı Example Surface cluster algebras Abstract Snake Graphs Relation to Cluster Algebras Self-crossing snake graphs Application G G 1 G 2 Notation • G = ( G 1 , G 2 , . . . , G d ) • G [ i , i + t ] = ( G i , G i +1 , . . . , G i + t ) • We denote by e i the interior edge between the tiles G i and G i +1 . ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 17 / 35

  60. On surface cluster algebras ˙ Ilke C ¸anak¸ cı Example Surface cluster algebras Abstract Snake Graphs Relation to Cluster Algebras Self-crossing snake graphs Application G G 1 G 2 Notation • G = ( G 1 , G 2 , . . . , G d ) • G [ i , i + t ] = ( G i , G i +1 , . . . , G i + t ) • We denote by e i the interior edge between the tiles G i and G i +1 . ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 17 / 35

  61. On surface cluster algebras ˙ Ilke C ¸anak¸ cı Example Surface cluster algebras Abstract Snake Graphs Relation to Cluster Algebras Self-crossing snake graphs Application G G 1 G 2 Notation • G = ( G 1 , G 2 , . . . , G d ) • G [ i , i + t ] = ( G i , G i +1 , . . . , G i + t ) • We denote by e i the interior edge between the tiles G i and G i +1 . ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 17 / 35

  62. On surface cluster algebras Local Overlaps ˙ Ilke C ¸anak¸ cı Surface cluster Definition algebras We say two snake graphs G 1 and G 2 have a local overlap G if G is a Abstract Snake Graphs maximal subgraph contained in both G 1 and G 2 . Relation to Notation: G ∼ = G 1 [ s , · · · , t ] ∼ = G 2 [ s ′ , · · · , t ′ ] . Cluster Algebras Self-crossing snake graphs Example Application G 1 G 2 Therefore G is a local overlap of G 1 and G 2 . • Note that two snake graphs may have several overlaps. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 18 / 35

  63. On surface cluster algebras Local Overlaps ˙ Ilke C ¸anak¸ cı Surface cluster Definition algebras We say two snake graphs G 1 and G 2 have a local overlap G if G is a Abstract Snake Graphs maximal subgraph contained in both G 1 and G 2 . Relation to Notation: G ∼ = G 1 [ s , · · · , t ] ∼ = G 2 [ s ′ , · · · , t ′ ] . Cluster Algebras Self-crossing snake graphs Example Application G 1 G 2 Therefore G is a local overlap of G 1 and G 2 . • Note that two snake graphs may have several overlaps. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 18 / 35

  64. On surface cluster algebras Local Overlaps ˙ Ilke C ¸anak¸ cı Surface cluster Definition algebras We say two snake graphs G 1 and G 2 have a local overlap G if G is a Abstract Snake Graphs maximal subgraph contained in both G 1 and G 2 . Relation to Notation: G ∼ = G 1 [ s , · · · , t ] ∼ = G 2 [ s ′ , · · · , t ′ ] . Cluster Algebras Self-crossing snake graphs Example Application G 1 G 2 Therefore G is a local overlap of G 1 and G 2 . • Note that two snake graphs may have several overlaps. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 18 / 35

  65. On surface cluster algebras Local Overlaps ˙ Ilke C ¸anak¸ cı Surface cluster Definition algebras We say two snake graphs G 1 and G 2 have a local overlap G if G is a Abstract Snake Graphs maximal subgraph contained in both G 1 and G 2 . Relation to Notation: G ∼ = G 1 [ s , · · · , t ] ∼ = G 2 [ s ′ , · · · , t ′ ] . Cluster Algebras Self-crossing snake graphs Example Application G 1 G 2 Therefore G is a local overlap of G 1 and G 2 . • Note that two snake graphs may have several overlaps. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 18 / 35

  66. On surface cluster algebras Local Overlaps ˙ Ilke C ¸anak¸ cı Surface cluster Definition algebras We say two snake graphs G 1 and G 2 have a local overlap G if G is a Abstract Snake Graphs maximal subgraph contained in both G 1 and G 2 . Relation to Notation: G ∼ = G 1 [ s , · · · , t ] ∼ = G 2 [ s ′ , · · · , t ′ ] . Cluster Algebras Self-crossing snake graphs Example Application G 1 G 2 Therefore G is a local overlap of G 1 and G 2 . • Note that two snake graphs may have several overlaps. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 18 / 35

  67. On surface cluster algebras Local Overlaps ˙ Ilke C ¸anak¸ cı Surface cluster Definition algebras We say two snake graphs G 1 and G 2 have a local overlap G if G is a Abstract Snake Graphs maximal subgraph contained in both G 1 and G 2 . Relation to Notation: G ∼ = G 1 [ s , · · · , t ] ∼ = G 2 [ s ′ , · · · , t ′ ] . Cluster Algebras Self-crossing snake graphs Example Application G G 1 G 2 Therefore G is a local overlap of G 1 and G 2 . • Note that two snake graphs may have several overlaps. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 18 / 35

  68. On surface cluster algebras Local Overlaps ˙ Ilke C ¸anak¸ cı Surface cluster Definition algebras We say two snake graphs G 1 and G 2 have a local overlap G if G is a Abstract Snake Graphs maximal subgraph contained in both G 1 and G 2 . Relation to Notation: G ∼ = G 1 [ s , · · · , t ] ∼ = G 2 [ s ′ , · · · , t ′ ] . Cluster Algebras Self-crossing snake graphs Example Application G G 1 G 2 Therefore G is a local overlap of G 1 and G 2 . • Note that two snake graphs may have several overlaps. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 18 / 35

  69. On surface cluster algebras Local Overlaps ˙ Ilke C ¸anak¸ cı Surface cluster Definition algebras We say two snake graphs G 1 and G 2 have a local overlap G if G is a Abstract Snake Graphs maximal subgraph contained in both G 1 and G 2 . Relation to Notation: G ∼ = G 1 [ s , · · · , t ] ∼ = G 2 [ s ′ , · · · , t ′ ] . Cluster Algebras Self-crossing snake graphs Example Application G 1 G 2 Therefore G is a local overlap of G 1 and G 2 . • Note that two snake graphs may have several overlaps. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 18 / 35

  70. On surface cluster algebras Local Overlaps ˙ Ilke C ¸anak¸ cı Surface cluster Definition algebras We say two snake graphs G 1 and G 2 have a local overlap G if G is a Abstract Snake Graphs maximal subgraph contained in both G 1 and G 2 . Relation to Notation: G ∼ = G 1 [ s , · · · , t ] ∼ = G 2 [ s ′ , · · · , t ′ ] . Cluster Algebras Self-crossing snake graphs Example Application G 1 G 2 Therefore G is a local overlap of G 1 and G 2 . • Note that two snake graphs may have several overlaps. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 18 / 35

  71. On surface cluster algebras Local Overlaps ˙ Ilke C ¸anak¸ cı Surface cluster Definition algebras We say two snake graphs G 1 and G 2 have a local overlap G if G is a Abstract Snake Graphs maximal subgraph contained in both G 1 and G 2 . Relation to Notation: G ∼ = G 1 [ s , · · · , t ] ∼ = G 2 [ s ′ , · · · , t ′ ] . Cluster Algebras Self-crossing snake graphs Example Application G 1 G 2 Therefore G is a local overlap of G 1 and G 2 . • Note that two snake graphs may have several overlaps. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 18 / 35

  72. On surface cluster algebras Local Overlaps ˙ Ilke C ¸anak¸ cı Surface cluster Definition algebras We say two snake graphs G 1 and G 2 have a local overlap G if G is a Abstract Snake Graphs maximal subgraph contained in both G 1 and G 2 . Relation to Notation: G ∼ = G 1 [ s , · · · , t ] ∼ = G 2 [ s ′ , · · · , t ′ ] . Cluster Algebras Self-crossing snake graphs Example Application G 2 G 1 Therefore G is a local overlap of G 1 and G 2 . • Note that two snake graphs may have several overlaps. ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 18 / 35

  73. On surface cluster algebras Sign Function ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Graphs Relation to Definition Cluster Algebras Self-crossing A sign function f on a snake graph G is a map f from the set of snake graphs edges of G to { + , −} such that on every tile in G the north and the Application west edge have the same sign, the south and the east edge have the same sign and the sign on the north edge is opposite to the sign on the south edge. Example A sign function on G 1 and G 2 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 19 / 35

  74. On surface cluster algebras Sign Function ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Graphs Relation to Definition Cluster Algebras Self-crossing A sign function f on a snake graph G is a map f from the set of snake graphs edges of G to { + , −} such that on every tile in G the north and the Application west edge have the same sign, the south and the east edge have the same sign and the sign on the north edge is opposite to the sign on the south edge. Example A sign function on G 1 and G 2 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 19 / 35

  75. On surface cluster algebras Sign Function ˙ Ilke C ¸anak¸ cı Surface cluster algebras Abstract Snake Definition Graphs Relation to A sign function f on a snake graph G is a map f from the set of Cluster Algebras edges of G to { + , −} such that on every tile in G the north and the Self-crossing snake graphs west edge have the same sign, the south and the east edge have the Application same sign and the sign on the north edge is opposite to the sign on the south edge. Example A sign function on G 1 and G 2 − + − + ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 19 / 35

  76. On surface cluster algebras Sign Function ˙ Ilke C ¸anak¸ cı Definition Surface cluster algebras A sign function f on a snake graph G is a map f from the set of Abstract Snake Graphs edges of G to { + , −} such that on every tile in G the north and the Relation to west edge have the same sign, the south and the east edge have the Cluster Algebras same sign and the sign on the north edge is opposite to the sign on Self-crossing snake graphs the south edge. Application Example A sign function on G 1 and G 2 − + − + G 1 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 19 / 35

  77. On surface cluster algebras Sign Function ˙ Ilke C ¸anak¸ cı Definition Surface cluster algebras A sign function f on a snake graph G is a map f from the set of Abstract Snake Graphs edges of G to { + , −} such that on every tile in G the north and the Relation to west edge have the same sign, the south and the east edge have the Cluster Algebras same sign and the sign on the north edge is opposite to the sign on Self-crossing snake graphs the south edge. Application Example A sign function on G 1 and G 2 − + − + + G 1 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 19 / 35

  78. On surface cluster algebras Sign Function ˙ Ilke C ¸anak¸ cı Definition Surface cluster algebras A sign function f on a snake graph G is a map f from the set of Abstract Snake Graphs edges of G to { + , −} such that on every tile in G the north and the Relation to west edge have the same sign, the south and the east edge have the Cluster Algebras same sign and the sign on the north edge is opposite to the sign on Self-crossing snake graphs the south edge. Application Example A sign function on G 1 and G 2 − + − + + + G 1 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 19 / 35

  79. On surface cluster algebras Sign Function ˙ Ilke C ¸anak¸ cı Definition Surface cluster algebras A sign function f on a snake graph G is a map f from the set of Abstract Snake Graphs edges of G to { + , −} such that on every tile in G the north and the Relation to west edge have the same sign, the south and the east edge have the Cluster Algebras same sign and the sign on the north edge is opposite to the sign on Self-crossing snake graphs the south edge. Application Example A sign function on G 1 and G 2 − + − + + − + G 1 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 19 / 35

  80. On surface cluster algebras Sign Function ˙ Ilke C ¸anak¸ cı Definition Surface cluster algebras A sign function f on a snake graph G is a map f from the set of Abstract Snake Graphs edges of G to { + , −} such that on every tile in G the north and the Relation to west edge have the same sign, the south and the east edge have the Cluster Algebras same sign and the sign on the north edge is opposite to the sign on Self-crossing snake graphs the south edge. Application Example A sign function on G 1 and G 2 ++ − + + + − + − − − − + + − −− + + −− −− + + − + + G 1 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 19 / 35

  81. On surface cluster algebras Sign Function ˙ Ilke C ¸anak¸ cı Definition Surface cluster algebras A sign function f on a snake graph G is a map f from the set of Abstract Snake Graphs edges of G to { + , −} such that on every tile in G the north and the Relation to west edge have the same sign, the south and the east edge have the Cluster Algebras same sign and the sign on the north edge is opposite to the sign on Self-crossing snake graphs the south edge. Application Example A sign function on G 1 and G 2 ++ − + + + − + − − − + −− + −− −− + + − + + G 1 G 2 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 19 / 35

  82. On surface cluster algebras Sign Function ˙ Ilke C ¸anak¸ cı Definition Surface cluster algebras A sign function f on a snake graph G is a map f from the set of Abstract Snake Graphs edges of G to { + , −} such that on every tile in G the north and the Relation to west edge have the same sign, the south and the east edge have the Cluster Algebras same sign and the sign on the north edge is opposite to the sign on Self-crossing snake graphs the south edge. Application Example A sign function on G 1 and G 2 ++ − + + + − + − − − + −− + −− −− + + + − + + G 1 G 2 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 19 / 35

  83. On surface cluster algebras Sign Function ˙ Ilke C ¸anak¸ cı Definition Surface cluster algebras A sign function f on a snake graph G is a map f from the set of Abstract Snake Graphs edges of G to { + , −} such that on every tile in G the north and the Relation to west edge have the same sign, the south and the east edge have the Cluster Algebras same sign and the sign on the north edge is opposite to the sign on Self-crossing snake graphs the south edge. Application Example A sign function on G 1 and G 2 ++ − + + + + − − − − + + − − − − − − + + − + −− + − −− − − −− + + + − + + G 1 G 2 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 19 / 35

  84. On surface cluster algebras Crossing ˙ Ilke C ¸anak¸ cı Definition Surface cluster algebras We say that G 1 and G 2 cross in a local overlap G if one of the Abstract Snake Graphs following conditions hold. Relation to Cluster Algebras • f 1 ( e s − 1 ) = − f 1 ( e t ) if s > 1 , t < d Self-crossing t ′ ) if s > 1 , t < d , s ′ = 1 , t ′ < d ′ • f 1 ( e s − 1 ) = f 2 ( e ′ snake graphs Application Example G 1 and G 2 cross at the overlap G . G 1 G 2 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 20 / 35

  85. On surface cluster algebras Crossing ˙ Ilke C ¸anak¸ cı Definition Surface cluster algebras We say that G 1 and G 2 cross in a local overlap G if one of the Abstract Snake Graphs following conditions hold. Relation to Cluster Algebras • f 1 ( e s − 1 ) = − f 1 ( e t ) if s > 1 , t < d Self-crossing t ′ ) if s > 1 , t < d , s ′ = 1 , t ′ < d ′ • f 1 ( e s − 1 ) = f 2 ( e ′ snake graphs Application Example G 1 and G 2 cross at the overlap G . G 1 G 2 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 20 / 35

  86. On surface cluster algebras Crossing ˙ Ilke C ¸anak¸ cı Definition Surface cluster algebras We say that G 1 and G 2 cross in a local overlap G if one of the Abstract Snake Graphs following conditions hold. Relation to Cluster Algebras • f 1 ( e s − 1 ) = − f 1 ( e t ) if s > 1 , t < d Self-crossing t ′ ) if s > 1 , t < d , s ′ = 1 , t ′ < d ′ • f 1 ( e s − 1 ) = f 2 ( e ′ snake graphs Application Example G 1 and G 2 cross at the overlap G . G 1 G 2 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 20 / 35

  87. On surface cluster algebras Crossing ˙ Ilke C ¸anak¸ cı Definition Surface cluster algebras We say that G 1 and G 2 cross in a local overlap G if one of the Abstract Snake Graphs following conditions hold. Relation to Cluster Algebras • f 1 ( e s − 1 ) = − f 1 ( e t ) if s > 1 , t < d Self-crossing t ′ ) if s > 1 , t < d , s ′ = 1 , t ′ < d ′ • f 1 ( e s − 1 ) = f 2 ( e ′ snake graphs Application Example G 1 and G 2 cross at the overlap G . G 1 G 2 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 20 / 35

  88. On surface cluster algebras Crossing ˙ Ilke C ¸anak¸ cı Definition Surface cluster algebras We say that G 1 and G 2 cross in a local overlap G if one of the Abstract Snake Graphs following conditions hold. Relation to Cluster Algebras • f 1 ( e s − 1 ) = − f 1 ( e t ) if s > 1 , t < d Self-crossing t ′ ) if s > 1 , t < d , s ′ = 1 , t ′ < d ′ • f 1 ( e s − 1 ) = f 2 ( e ′ snake graphs Application Example G 1 and G 2 cross at the overlap G . + G 1 G 2 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 20 / 35

  89. On surface cluster algebras Crossing ˙ Ilke C ¸anak¸ cı Definition Surface cluster algebras We say that G 1 and G 2 cross in a local overlap G if one of the Abstract Snake Graphs following conditions hold. Relation to Cluster Algebras • f 1 ( e s − 1 ) = − f 1 ( e t ) if s > 1 , t < d Self-crossing t ′ ) if s > 1 , t < d , s ′ = 1 , t ′ < d ′ • f 1 ( e s − 1 ) = f 2 ( e ′ snake graphs Application Example G 1 and G 2 cross at the overlap G . − + G 1 G 2 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 20 / 35

  90. On surface cluster algebras Crossing ˙ Ilke C ¸anak¸ cı Definition Surface cluster algebras We say that G 1 and G 2 cross in a local overlap G if one of the Abstract Snake Graphs following conditions hold. Relation to Cluster Algebras • f 1 ( e s − 1 ) = − f 1 ( e t ) if s > 1 , t < d Self-crossing t ′ ) if s > 1 , t < d , s ′ = 1 , t ′ < d ′ • f 1 ( e s − 1 ) = f 2 ( e ′ snake graphs Application Example G 1 and G 2 cross at the overlap G . − + G 1 G 2 ˙ Ilke C ¸anak¸ cı (U. Leicester) On surface cluster algebras Geometry Seminar (U. Bath) 20 / 35

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