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Band and snake graph calculus I. Canakci, R. Schiffler Motivation On surface cluster algebras: Band and snake Abstract Snake Graphs Relation to graph calculus Cluster Algebras Band Graphs and Future Directions Ilke Canakci 1 Ralf

  1. Band and snake graph calculus Background I. Canakci, R. Schiffler Motivation • Cluster algebras, introduced by Fomin and Zelevinsky in [FZ1] Abstract Snake form a class of combinatorially defined commutative algebras, Graphs and the set of generators of a cluster algebra, cluster Relation to Cluster Algebras variables , is obtained by an iterative process. Band Graphs and Future Directions • A surface cluster algebra A ( S , M ) is associated to a surface S with boundary that has finitely many marked points. γ 5 γ 4 2 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] associates a connected graph, called the snake graph to each arc in the surface to obtain a direct formula for cluster variables of surface cluster algebras. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 2 / 18

  2. Band and snake graph calculus Motivation I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Let A ( S , M ) cluster algebra associated to a surface ( S , M ) . Relation to Cluster Algebras Band Graphs and Future Directions We have the following situation: 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? I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 3 / 18

  3. Band and snake graph calculus Motivation I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Let A ( S , M ) cluster algebra associated to a surface ( S , M ) . Relation to Cluster Algebras Band Graphs and We have the following situation: Future Directions [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? I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 3 / 18

  4. Band and snake graph calculus Motivation I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Let A ( S , M ) cluster algebra associated to a surface ( S , M ) . Relation to Cluster Algebras Band Graphs and We have the following situation: Future Directions [FST] [MSW] ← → arc − → cluster variable snake graph 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? I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 3 / 18

  5. Band and snake graph calculus Motivation I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Let A ( S , M ) cluster algebra associated to a surface ( S , M ) . Relation to Cluster Algebras Band Graphs and We have the following situation: Future Directions [FST] [MSW] ← → arc − → cluster variable snake graph 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? I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 3 / 18

  6. Band and snake graph calculus Motivation I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Let A ( S , M ) cluster algebra associated to a surface ( S , M ) . Relation to Cluster Algebras Band Graphs and We have the following situation: Future Directions [FST] [MSW] ← → arc − → cluster variable snake graph 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? I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 3 / 18

  7. Band and snake graph calculus Motivation I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Let A ( S , M ) cluster algebra associated to a surface ( S , M ) . Relation to Cluster Algebras Band Graphs and We have the following situation: Future Directions [FST] [MSW] ← → arc − → cluster variable snake graph 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? I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 3 / 18

  8. Band and snake graph calculus Motivation I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Let A ( S , M ) cluster algebra associated to a surface ( S , M ) . Relation to Cluster Algebras Band Graphs and We have the following situation: Future Directions [FST] [MSW] ← → arc − → cluster variable snake graph 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? I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 3 / 18

  9. Band and snake graph calculus Abstract Snake Graphs I. Canakci, R. Schiffler Definition Motivation 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 Band Graphs and (i) G i and G i +1 share exactly one edge e i and this edge is either the Future Directions north edge of G i and the south edge of G i +1 or the east edge of 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 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 4 / 18

  10. Band and snake graph calculus Abstract Snake Graphs I. Canakci, R. Schiffler Definition Motivation 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 Band Graphs and (i) G i and G i +1 share exactly one edge e i and this edge is either the Future Directions north edge of G i and the south edge of G i +1 or the east edge of 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 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 4 / 18

  11. Band and snake graph calculus Abstract Snake Graphs I. Canakci, R. Schiffler Definition Motivation 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 Band Graphs and (i) G i and G i +1 share exactly one edge e i and this edge is either the Future Directions north edge of G i and the south edge of G i +1 or the east edge of 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 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 4 / 18

  12. Band and snake graph calculus I. Canakci, R. Schiffler Example Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions 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 . I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 5 / 18

  13. Band and snake graph calculus I. Canakci, R. Schiffler Example Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions 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 . I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 5 / 18

  14. Band and snake graph calculus I. Canakci, R. Schiffler Example Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions 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 . I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 5 / 18

  15. Band and snake graph calculus I. Canakci, R. Schiffler Example Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions 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 . I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 5 / 18

  16. Band and snake graph calculus I. Canakci, R. Schiffler Example Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions 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 . I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 5 / 18

  17. Band and snake graph calculus Local Overlaps I. Canakci, R. Schiffler Definition Motivation 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 Cluster Algebras Notation: G ∼ = G 1 [ s , · · · , t ] ∼ = G 2 [ s ′ , · · · , t ′ ] . Band Graphs and Future Directions Example 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. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 6 / 18

  18. Band and snake graph calculus Local Overlaps I. Canakci, R. Schiffler Definition Motivation 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 Cluster Algebras Notation: G ∼ = G 1 [ s , · · · , t ] ∼ = G 2 [ s ′ , · · · , t ′ ] . Band Graphs and Future Directions Example 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. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 6 / 18

  19. Band and snake graph calculus Local Overlaps I. Canakci, R. Schiffler Definition Motivation 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 Cluster Algebras Notation: G ∼ = G 1 [ s , · · · , t ] ∼ = G 2 [ s ′ , · · · , t ′ ] . Band Graphs and Future Directions Example 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. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 6 / 18

  20. Band and snake graph calculus Local Overlaps I. Canakci, R. Schiffler Definition Motivation 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 Cluster Algebras Notation: G ∼ = G 1 [ s , · · · , t ] ∼ = G 2 [ s ′ , · · · , t ′ ] . Band Graphs and Future Directions Example 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. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 6 / 18

  21. Band and snake graph calculus Local Overlaps I. Canakci, R. Schiffler Definition Motivation 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 Cluster Algebras Notation: G ∼ = G 1 [ s , · · · , t ] ∼ = G 2 [ s ′ , · · · , t ′ ] . Band Graphs and Future Directions Example 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. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 6 / 18

  22. Band and snake graph calculus Local Overlaps I. Canakci, R. Schiffler Definition Motivation 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 Cluster Algebras Notation: G ∼ = G 1 [ s , · · · , t ] ∼ = G 2 [ s ′ , · · · , t ′ ] . Band Graphs and Future Directions Example 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. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 6 / 18

  23. Band and snake graph calculus Local Overlaps I. Canakci, R. Schiffler Definition Motivation 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 Cluster Algebras Notation: G ∼ = G 1 [ s , · · · , t ] ∼ = G 2 [ s ′ , · · · , t ′ ] . Band Graphs and Future Directions Example 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. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 6 / 18

  24. Band and snake graph calculus Local Overlaps I. Canakci, R. Schiffler Definition Motivation 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 Cluster Algebras Notation: G ∼ = G 1 [ s , · · · , t ] ∼ = G 2 [ s ′ , · · · , t ′ ] . Band Graphs and Future Directions Example 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. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 6 / 18

  25. Band and snake graph calculus Sign Function I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Relation to Cluster Algebras Definition Band Graphs and A sign function f on a snake graph G is a map f from the set of Future Directions edges of G to { + , −} such that on every tile in G the north and the 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 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

  26. Band and snake graph calculus Sign Function I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Relation to Cluster Algebras Definition Band Graphs and A sign function f on a snake graph G is a map f from the set of Future Directions edges of G to { + , −} such that on every tile in G the north and the 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 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

  27. Band and snake graph calculus Sign Function I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Definition 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 Band Graphs and Future Directions 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 − + − + I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

  28. Band and snake graph calculus Sign Function I. Canakci, R. Schiffler Definition Motivation 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 Band Graphs and Future Directions the south edge. Example A sign function on G 1 and G 2 − + − + G 1 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

  29. Band and snake graph calculus Sign Function I. Canakci, R. Schiffler Definition Motivation 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 Band Graphs and Future Directions the south edge. Example A sign function on G 1 and G 2 − + − + + G 1 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

  30. Band and snake graph calculus Sign Function I. Canakci, R. Schiffler Definition Motivation 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 Band Graphs and Future Directions the south edge. Example A sign function on G 1 and G 2 − + − + + + G 1 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

  31. Band and snake graph calculus Sign Function I. Canakci, R. Schiffler Definition Motivation 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 Band Graphs and Future Directions the south edge. Example A sign function on G 1 and G 2 − + − + + − + G 1 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

  32. Band and snake graph calculus Sign Function I. Canakci, R. Schiffler Definition Motivation 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 Band Graphs and Future Directions the south edge. Example A sign function on G 1 and G 2 ++ − + + + − + − − − − + + − −− + + −− −− + + − + + G 1 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

  33. Band and snake graph calculus Sign Function I. Canakci, R. Schiffler Definition Motivation 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 Band Graphs and Future Directions the south edge. Example A sign function on G 1 and G 2 ++ − + + + − + − − − + −− + −− −− + + − + + G 1 G 2 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

  34. Band and snake graph calculus Sign Function I. Canakci, R. Schiffler Definition Motivation 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 Band Graphs and Future Directions the south edge. Example A sign function on G 1 and G 2 ++ − + + + − + − − − + −− + −− −− + + + − + + G 1 G 2 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

  35. Band and snake graph calculus Sign Function I. Canakci, R. Schiffler Definition Motivation 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 Band Graphs and Future Directions the south edge. Example A sign function on G 1 and G 2 ++ − + + + + − − − − + + − − − − − − + + − + −− + − −− − − −− + + + − + + G 1 G 2 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

  36. Band and snake graph calculus Crossing I. Canakci, R. Schiffler Definition Motivation 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 Band Graphs and t ′ ) if s > 1 , t < d , s ′ = 1 , t ′ < d ′ Future Directions • f 1 ( e s − 1 ) = f 2 ( e ′ Example G 1 and G 2 cross at the overlap G . G 1 G 2 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 8 / 18

  37. Band and snake graph calculus Crossing I. Canakci, R. Schiffler Definition Motivation 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 Band Graphs and t ′ ) if s > 1 , t < d , s ′ = 1 , t ′ < d ′ Future Directions • f 1 ( e s − 1 ) = f 2 ( e ′ Example G 1 and G 2 cross at the overlap G . G 1 G 2 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 8 / 18

  38. Band and snake graph calculus Crossing I. Canakci, R. Schiffler Definition Motivation 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 Band Graphs and t ′ ) if s > 1 , t < d , s ′ = 1 , t ′ < d ′ Future Directions • f 1 ( e s − 1 ) = f 2 ( e ′ Example G 1 and G 2 cross at the overlap G . G 1 G 2 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 8 / 18

  39. Band and snake graph calculus Crossing I. Canakci, R. Schiffler Definition Motivation 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 Band Graphs and t ′ ) if s > 1 , t < d , s ′ = 1 , t ′ < d ′ Future Directions • f 1 ( e s − 1 ) = f 2 ( e ′ Example G 1 and G 2 cross at the overlap G . G 1 G 2 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 8 / 18

  40. Band and snake graph calculus Crossing I. Canakci, R. Schiffler Definition Motivation 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 Band Graphs and t ′ ) if s > 1 , t < d , s ′ = 1 , t ′ < d ′ Future Directions • f 1 ( e s − 1 ) = f 2 ( e ′ Example G 1 and G 2 cross at the overlap G . + G 1 G 2 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 8 / 18

  41. Band and snake graph calculus Crossing I. Canakci, R. Schiffler Definition Motivation 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 Band Graphs and t ′ ) if s > 1 , t < d , s ′ = 1 , t ′ < d ′ Future Directions • f 1 ( e s − 1 ) = f 2 ( e ′ Example G 1 and G 2 cross at the overlap G . − + G 1 G 2 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 8 / 18

  42. Band and snake graph calculus Crossing I. Canakci, R. Schiffler Definition Motivation 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 Band Graphs and t ′ ) if s > 1 , t < d , s ′ = 1 , t ′ < d ′ Future Directions • f 1 ( e s − 1 ) = f 2 ( e ′ Example G 1 and G 2 cross at the overlap G . − + G 1 G 2 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 8 / 18

  43. Band and snake graph calculus Crossing I. Canakci, R. Schiffler Definition Motivation 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 Band Graphs and t ′ ) if s > 1 , t < d , s ′ = 1 , t ′ < d ′ Future Directions • f 1 ( e s − 1 ) = f 2 ( e ′ Example G 1 and G 2 cross at the overlap G . − + + G 1 G 2 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 8 / 18

  44. Band and snake graph calculus Example: Resolution Res G ( G 1 , G 2 ) I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions G 2 G 1 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 9 / 18

  45. Band and snake graph calculus Example: Resolution Res G ( G 1 , G 2 ) I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions G 1 G 2 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 9 / 18

  46. Band and snake graph calculus Example: Resolution Res G ( G 1 , G 2 ) I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions G 1 G 2 G 3 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 9 / 18

  47. Band and snake graph calculus Example: Resolution Res G ( G 1 , G 2 ) I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions G 1 G 2 G 3 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 9 / 18

  48. Band and snake graph calculus Example: Resolution Res G ( G 1 , G 2 ) I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions G 1 G 2 G 4 G 3 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 9 / 18

  49. Band and snake graph calculus Example: Resolution Res G ( G 1 , G 2 ) I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions G 1 G 2 G 4 G 3 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 9 / 18

  50. Band and snake graph calculus Example: Resolution Res G ( G 1 , G 2 ) I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions + G 2 G 1 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 9 / 18

  51. Band and snake graph calculus Example: Resolution Res G ( G 1 , G 2 ) I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions + − − − − − − G 2 G 1 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 9 / 18

  52. Band and snake graph calculus Example: Resolution Res G ( G 1 , G 2 ) I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions + − − − − − − + G 2 G 1 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 9 / 18

  53. Band and snake graph calculus Example: Resolution Res G ( G 1 , G 2 ) I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions + − − − − − − + G 2 G 1 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 9 / 18

  54. Band and snake graph calculus Example: Resolution (Continued) I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions G 1 G 2 G 5 G 4 G 3 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 10 / 18

  55. Band and snake graph calculus Example: Resolution (Continued) I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions G 1 G 2 G 5 G 6 G 4 G 3 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 10 / 18

  56. Band and snake graph calculus Example: Resolution (Continued) I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions G 1 G 2 G 5 G 6 G 4 G 3 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 10 / 18

  57. Band and snake graph calculus Resolution: Definition I. Canakci, R. Schiffler Motivation Assumption: We will assume that s > 1 , t < d , s ′ = 1 and t ′ < d ′ . Abstract Snake For all other cases, see [CS]. Graphs Relation to Cluster Algebras We define four connected subgraphs as follows. Band Graphs and • G 3 = G 1 [1 , t ] ∪ G 2 [ t ′ + 1 , d ′ ] , Future Directions • G 4 = G 2 [1 , t ′ ] ∪ G 1 [ t + 1 , d ] , • G 5 = G 1 [1 , k ] where k < s − 1 is the largest integer such that the sign on the interior edge between tiles k and k + 1 is the same as the sign on the interior edge of tiles s − 1 and s , • G 6 = G 2 [ d ′ , t ′ + 1] ∪ G 1 [ t + 1 , d ] where the two subgraphs are glued along the south G t +1 and the north of G ′ t ′ +1 if G t +1 is north of G t in G 1 . Definition The resolution of the crossing of G 1 and G 2 in G is defined to be ( G 3 ⊔ G 4 , G 5 ⊔ G 6 ) and is denoted by Res G ( G 1 , G 2 ) . I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 11 / 18

  58. Band and snake graph calculus Resolution: Definition I. Canakci, R. Schiffler Motivation Assumption: We will assume that s > 1 , t < d , s ′ = 1 and t ′ < d ′ . Abstract Snake For all other cases, see [CS]. Graphs Relation to Cluster Algebras We define four connected subgraphs as follows. Band Graphs and • G 3 = G 1 [1 , t ] ∪ G 2 [ t ′ + 1 , d ′ ] , Future Directions • G 4 = G 2 [1 , t ′ ] ∪ G 1 [ t + 1 , d ] , • G 5 = G 1 [1 , k ] where k < s − 1 is the largest integer such that the sign on the interior edge between tiles k and k + 1 is the same as the sign on the interior edge of tiles s − 1 and s , • G 6 = G 2 [ d ′ , t ′ + 1] ∪ G 1 [ t + 1 , d ] where the two subgraphs are glued along the south G t +1 and the north of G ′ t ′ +1 if G t +1 is north of G t in G 1 . Definition The resolution of the crossing of G 1 and G 2 in G is defined to be ( G 3 ⊔ G 4 , G 5 ⊔ G 6 ) and is denoted by Res G ( G 1 , G 2 ) . I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 11 / 18

  59. Band and snake graph calculus Resolution: Definition I. Canakci, R. Schiffler Motivation Assumption: We will assume that s > 1 , t < d , s ′ = 1 and t ′ < d ′ . Abstract Snake For all other cases, see [CS]. Graphs Relation to Cluster Algebras We define four connected subgraphs as follows. Band Graphs and • G 3 = G 1 [1 , t ] ∪ G 2 [ t ′ + 1 , d ′ ] , Future Directions • G 4 = G 2 [1 , t ′ ] ∪ G 1 [ t + 1 , d ] , • G 5 = G 1 [1 , k ] where k < s − 1 is the largest integer such that the sign on the interior edge between tiles k and k + 1 is the same as the sign on the interior edge of tiles s − 1 and s , • G 6 = G 2 [ d ′ , t ′ + 1] ∪ G 1 [ t + 1 , d ] where the two subgraphs are glued along the south G t +1 and the north of G ′ t ′ +1 if G t +1 is north of G t in G 1 . Definition The resolution of the crossing of G 1 and G 2 in G is defined to be ( G 3 ⊔ G 4 , G 5 ⊔ G 6 ) and is denoted by Res G ( G 1 , G 2 ) . I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 11 / 18

  60. Band and snake graph calculus Resolution: Definition I. Canakci, R. Schiffler Motivation Assumption: We will assume that s > 1 , t < d , s ′ = 1 and t ′ < d ′ . Abstract Snake For all other cases, see [CS]. Graphs Relation to Cluster Algebras We define four connected subgraphs as follows. Band Graphs and • G 3 = G 1 [1 , t ] ∪ G 2 [ t ′ + 1 , d ′ ] , Future Directions • G 4 = G 2 [1 , t ′ ] ∪ G 1 [ t + 1 , d ] , • G 5 = G 1 [1 , k ] where k < s − 1 is the largest integer such that the sign on the interior edge between tiles k and k + 1 is the same as the sign on the interior edge of tiles s − 1 and s , • G 6 = G 2 [ d ′ , t ′ + 1] ∪ G 1 [ t + 1 , d ] where the two subgraphs are glued along the south G t +1 and the north of G ′ t ′ +1 if G t +1 is north of G t in G 1 . Definition The resolution of the crossing of G 1 and G 2 in G is defined to be ( G 3 ⊔ G 4 , G 5 ⊔ G 6 ) and is denoted by Res G ( G 1 , G 2 ) . I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 11 / 18

  61. Band and snake graph calculus Resolution: Definition I. Canakci, R. Schiffler Motivation Assumption: We will assume that s > 1 , t < d , s ′ = 1 and t ′ < d ′ . Abstract Snake For all other cases, see [CS]. Graphs Relation to Cluster Algebras We define four connected subgraphs as follows. Band Graphs and • G 3 = G 1 [1 , t ] ∪ G 2 [ t ′ + 1 , d ′ ] , Future Directions • G 4 = G 2 [1 , t ′ ] ∪ G 1 [ t + 1 , d ] , • G 5 = G 1 [1 , k ] where k < s − 1 is the largest integer such that the sign on the interior edge between tiles k and k + 1 is the same as the sign on the interior edge of tiles s − 1 and s , • G 6 = G 2 [ d ′ , t ′ + 1] ∪ G 1 [ t + 1 , d ] where the two subgraphs are glued along the south G t +1 and the north of G ′ t ′ +1 if G t +1 is north of G t in G 1 . Definition The resolution of the crossing of G 1 and G 2 in G is defined to be ( G 3 ⊔ G 4 , G 5 ⊔ G 6 ) and is denoted by Res G ( G 1 , G 2 ) . I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 11 / 18

  62. Band and snake graph calculus Bijection of Perfect Matchings I. Canakci, R. Schiffler Definition Motivation A perfect matching P of a graph G is a subset of the set of edges Abstract Snake Graphs of G such that each vertex of G is incident to exactly one edge in P . Relation to Cluster Algebras Band Graphs and Future Directions • Let Match ( G ) denote the set of all perfect matchings of the graph G and Match (Res G ( G 1 , G 2 )) = Match ( G 3 ⊔ G 4 ) ∪ Match ( G 5 ⊔ G 6 ) . Theorem (CS) Let G 1 , G 2 be two snake graphs. Then there is a bijection Match ( G 1 ⊔ G 2 ) − → Match (Res G ( G 1 , G 2 )) • Note that we construct the bijection map and its inverse map explicitly. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 12 / 18

  63. Band and snake graph calculus Bijection of Perfect Matchings I. Canakci, R. Schiffler Definition Motivation A perfect matching P of a graph G is a subset of the set of edges Abstract Snake Graphs of G such that each vertex of G is incident to exactly one edge in P . Relation to Cluster Algebras Band Graphs and Future Directions • Let Match ( G ) denote the set of all perfect matchings of the graph G and Match (Res G ( G 1 , G 2 )) = Match ( G 3 ⊔ G 4 ) ∪ Match ( G 5 ⊔ G 6 ) . Theorem (CS) Let G 1 , G 2 be two snake graphs. Then there is a bijection Match ( G 1 ⊔ G 2 ) − → Match (Res G ( G 1 , G 2 )) • Note that we construct the bijection map and its inverse map explicitly. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 12 / 18

  64. Band and snake graph calculus Bijection of Perfect Matchings I. Canakci, R. Schiffler Definition Motivation A perfect matching P of a graph G is a subset of the set of edges Abstract Snake Graphs of G such that each vertex of G is incident to exactly one edge in P . Relation to Cluster Algebras Band Graphs and Future Directions • Let Match ( G ) denote the set of all perfect matchings of the graph G and Match (Res G ( G 1 , G 2 )) = Match ( G 3 ⊔ G 4 ) ∪ Match ( G 5 ⊔ G 6 ) . Theorem (CS) Let G 1 , G 2 be two snake graphs. Then there is a bijection Match ( G 1 ⊔ G 2 ) − → Match (Res G ( G 1 , G 2 )) • Note that we construct the bijection map and its inverse map explicitly. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 12 / 18

  65. Band and snake graph calculus Bijection of Perfect Matchings I. Canakci, R. Schiffler Definition Motivation A perfect matching P of a graph G is a subset of the set of edges Abstract Snake Graphs of G such that each vertex of G is incident to exactly one edge in P . Relation to Cluster Algebras Band Graphs and Future Directions • Let Match ( G ) denote the set of all perfect matchings of the graph G and Match (Res G ( G 1 , G 2 )) = Match ( G 3 ⊔ G 4 ) ∪ Match ( G 5 ⊔ G 6 ) . Theorem (CS) Let G 1 , G 2 be two snake graphs. Then there is a bijection Match ( G 1 ⊔ G 2 ) − → Match (Res G ( G 1 , G 2 )) • Note that we construct the bijection map and its inverse map explicitly. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 12 / 18

  66. Band and snake graph calculus Bijection of Perfect Matchings I. Canakci, R. Schiffler Definition Motivation A perfect matching P of a graph G is a subset of the set of edges Abstract Snake Graphs of G such that each vertex of G is incident to exactly one edge in P . Relation to Cluster Algebras Band Graphs and Future Directions • Let Match ( G ) denote the set of all perfect matchings of the graph G and Match (Res G ( G 1 , G 2 )) = Match ( G 3 ⊔ G 4 ) ∪ Match ( G 5 ⊔ G 6 ) . Theorem (CS) Let G 1 , G 2 be two snake graphs. Then there is a bijection Match ( G 1 ⊔ G 2 ) − → Match (Res G ( G 1 , G 2 )) • Note that we construct the bijection map and its inverse map explicitly. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 12 / 18

  67. Band and snake graph calculus Bijection of Perfect Matchings I. Canakci, R. Schiffler Definition Motivation A perfect matching P of a graph G is a subset of the set of edges Abstract Snake Graphs of G such that each vertex of G is incident to exactly one edge in P . Relation to Cluster Algebras Band Graphs and Future Directions • Let Match ( G ) denote the set of all perfect matchings of the graph G and Match (Res G ( G 1 , G 2 )) = Match ( G 3 ⊔ G 4 ) ∪ Match ( G 5 ⊔ G 6 ) . Theorem (CS) Let G 1 , G 2 be two snake graphs. Then there is a bijection Match ( G 1 ⊔ G 2 ) − → Match (Res G ( G 1 , G 2 )) • Note that we construct the bijection map and its inverse map explicitly. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 12 / 18

  68. Band and snake graph calculus Surface Example I. Canakci, R. Schiffler Motivation 1 Abstract Snake 2 Graphs 3 Relation to 4 Cluster Algebras 5 6 7 Band Graphs and 8 Future Directions 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

  69. Band and snake graph calculus Surface Example I. Canakci, R. Schiffler Motivation 1 Abstract Snake 2 Graphs 3 Relation to 4 Cluster Algebras 5 6 7 Band Graphs and 8 Future Directions 9 10 γ 1 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

  70. Band and snake graph calculus Surface Example I. Canakci, R. Schiffler Motivation 1 Abstract Snake 2 Graphs 3 Relation to 4 Cluster Algebras 5 6 7 Band Graphs and 8 Future Directions 9 10 γ 1 11 12 13 14 15 16 17 18 γ 2 26 19 27 28 20 29 21 30 22 31 23 24 32 33 25 34 35 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

  71. Band and snake graph calculus Surface Example I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Relation to 22 23 24 25 35 Cluster Algebras 1 20 21 34 2 19 32 33 Band Graphs and 3 18 31 Future Directions 4 17 30 5 6 15 16 28 29 7 14 18 26 27 8 9 11 12 13 17 10 γ 1 9 10 15 16 11 12 7 8 14 6 13 13 2 3 4 5 14 G 1 G 2 1 15 16 17 18 γ 2 26 19 27 28 20 29 21 30 22 31 23 24 32 33 25 34 35 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

  72. Band and snake graph calculus Surface Example I. Canakci, R. Schiffler Motivation 1 Abstract Snake 2 Graphs 3 Relation to 4 Cluster Algebras 5 6 7 Band Graphs and 8 Future Directions 9 10 γ 1 11 12 13 14 15 16 17 18 γ 2 26 19 27 28 20 29 21 30 22 31 23 24 32 33 25 34 35 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

  73. Band and snake graph calculus Surface Example I. Canakci, R. Schiffler Motivation 1 Abstract Snake 2 Graphs 3 Relation to 4 Cluster Algebras 5 6 7 Band Graphs and 8 Future Directions 9 10 γ 1 11 12 γ 3 13 14 15 16 17 18 γ 2 26 19 27 28 20 29 21 30 22 31 23 24 32 33 25 34 35 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

  74. Band and snake graph calculus Surface Example I. Canakci, R. Schiffler Motivation 1 Abstract Snake 2 Graphs 3 Relation to 4 Cluster Algebras 5 6 7 Band Graphs and 8 Future Directions 9 10 γ 1 11 12 γ 3 γ 4 13 14 15 16 17 18 γ 2 26 19 27 28 20 29 21 30 22 31 23 24 32 33 25 34 35 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

  75. Band and snake graph calculus Surface Example I. Canakci, R. Schiffler Motivation 1 Abstract Snake 2 Graphs 3 γ 5 Relation to 4 Cluster Algebras 5 6 7 Band Graphs and 8 Future Directions 9 10 γ 1 11 12 γ 3 γ 4 13 14 15 16 17 18 γ 2 26 19 27 28 20 29 21 30 22 31 23 24 32 33 25 34 35 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

  76. Band and snake graph calculus Surface Example I. Canakci, R. Schiffler Motivation 1 Abstract Snake 2 Graphs 3 γ 5 Relation to 4 Cluster Algebras 5 6 7 Band Graphs and 8 Future Directions 9 10 γ 1 11 12 γ 3 γ 4 13 14 15 16 17 18 γ 2 γ 6 26 19 27 28 20 29 21 30 22 31 23 24 32 33 25 34 35 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

  77. Band and snake graph calculus Surface Example I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Relation to 35 Cluster Algebras 34 Band Graphs and 32 33 22 23 24 25 Future Directions 31 22 23 24 25 20 21 30 20 21 19 28 29 19 27 26 18 26 27 18 29 28 2 3 4 5 17 17 30 1 15 16 15 16 31 14 14 33 32 11 12 13 13 G 5 34 9 10 35 7 8 6 2 3 4 5 G 4 G 6 1 G 3 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

  78. Band and snake graph calculus Surface Example I. Canakci, R. Schiffler Motivation Abstract Snake Graphs 22 23 24 25 35 1 20 21 34 2 Relation to 19 32 33 3 γ 5 Cluster Algebras 18 31 4 17 30 5 Band Graphs and 6 15 16 28 29 7 Future Directions 14 18 26 27 8 9 11 12 13 17 10 γ 1 9 10 15 16 11 12 7 8 14 γ 3 γ 4 6 13 13 2 3 4 5 14 G 1 G 2 1 15 16 17 18 35 γ 2 34 2 3 4 5 32 33 γ 6 26 19 1 27 31 28 20 30 22 23 24 25 29 G 5 21 28 29 20 21 30 22 31 18 26 27 19 23 24 32 17 18 22 23 24 25 33 25 15 16 17 34 20 21 14 15 16 19 35 11 12 13 14 27 26 9 10 13 G 4 29 28 7 8 30 6 31 2 3 4 5 33 32 1 G 3 34 35 G 6 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

  79. Band and snake graph calculus Surface Example I. Canakci, R. Schiffler Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions G 1 G 2 G 5 G 6 G 4 G 3 I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

  80. Band and snake graph calculus Relation to Cluster Algebras I. Canakci, R. Schiffler Motivation Abstract Snake Let γ 1 and γ 2 be two arcs and G 1 and G 2 their corresponding snake Graphs graphs. Relation to Cluster Algebras Theorem (CS) Band Graphs and Future Directions γ 1 and γ 2 cross if and only if G 1 and G 2 cross. Theorem (CS) If γ 1 and γ 2 cross, then the snake graphs of the four arcs obtained by smoothing the crossing are given by the resolution Res G ( G 1 , G 2 ) of the crossing of the snake graphs G 1 and G 2 at the overlap G . Remark We do not assume that γ 1 and γ 2 cross only once. If the arcs cross multiple times the theorem can be used to resolve any of the crossings. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 14 / 18

  81. Band and snake graph calculus Relation to Cluster Algebras I. Canakci, R. Schiffler Motivation Abstract Snake Let γ 1 and γ 2 be two arcs and G 1 and G 2 their corresponding snake Graphs graphs. Relation to Cluster Algebras Theorem (CS) Band Graphs and Future Directions γ 1 and γ 2 cross if and only if G 1 and G 2 cross. Theorem (CS) If γ 1 and γ 2 cross, then the snake graphs of the four arcs obtained by smoothing the crossing are given by the resolution Res G ( G 1 , G 2 ) of the crossing of the snake graphs G 1 and G 2 at the overlap G . Remark We do not assume that γ 1 and γ 2 cross only once. If the arcs cross multiple times the theorem can be used to resolve any of the crossings. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 14 / 18

  82. Band and snake graph calculus Relation to Cluster Algebras I. Canakci, R. Schiffler Motivation Abstract Snake Let γ 1 and γ 2 be two arcs and G 1 and G 2 their corresponding snake Graphs graphs. Relation to Cluster Algebras Theorem (CS) Band Graphs and Future Directions γ 1 and γ 2 cross if and only if G 1 and G 2 cross. Theorem (CS) If γ 1 and γ 2 cross, then the snake graphs of the four arcs obtained by smoothing the crossing are given by the resolution Res G ( G 1 , G 2 ) of the crossing of the snake graphs G 1 and G 2 at the overlap G . Remark We do not assume that γ 1 and γ 2 cross only once. If the arcs cross multiple times the theorem can be used to resolve any of the crossings. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 14 / 18

  83. Band and snake graph calculus Relation to Cluster Algebras I. Canakci, R. Schiffler Motivation Abstract Snake Let γ 1 and γ 2 be two arcs and G 1 and G 2 their corresponding snake Graphs graphs. Relation to Cluster Algebras Theorem (CS) Band Graphs and Future Directions γ 1 and γ 2 cross if and only if G 1 and G 2 cross. Theorem (CS) If γ 1 and γ 2 cross, then the snake graphs of the four arcs obtained by smoothing the crossing are given by the resolution Res G ( G 1 , G 2 ) of the crossing of the snake graphs G 1 and G 2 at the overlap G . Remark We do not assume that γ 1 and γ 2 cross only once. If the arcs cross multiple times the theorem can be used to resolve any of the crossings. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 14 / 18

  84. Band and snake graph calculus Skein Relations I. Canakci, R. Schiffler Motivation As a corollary we obtain a new proof of the skein relations [MW]. Abstract Snake Graphs Corollary (CS) Relation to Cluster Algebras Let γ 1 and γ 2 be two arcs which cross and let ( γ 3 , γ 4 ) and ( γ 5 , γ 6 ) be Band Graphs and the two pairs of arcs obtained by smoothing the crossing. Then Future Directions x γ 1 x γ 2 = x γ 3 x γ 4 + y ( ˜ G ) x γ 5 x γ 6 where ˜ G is the closure of the overlap G . Remark • Note that Musiker and Williams in [MW] use hyperbolic geometry to prove the skein relations. • Our proof is purely combinatorial . The key ingredient to our proof is Theorem 12 where we show the bijection between the perfect matchings. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 15 / 18

  85. Band and snake graph calculus Skein Relations I. Canakci, R. Schiffler Motivation As a corollary we obtain a new proof of the skein relations [MW]. Abstract Snake Graphs Corollary (CS) Relation to Cluster Algebras Let γ 1 and γ 2 be two arcs which cross and let ( γ 3 , γ 4 ) and ( γ 5 , γ 6 ) be Band Graphs and the two pairs of arcs obtained by smoothing the crossing. Then Future Directions x γ 1 x γ 2 = x γ 3 x γ 4 + y ( ˜ G ) x γ 5 x γ 6 where ˜ G is the closure of the overlap G . Remark • Note that Musiker and Williams in [MW] use hyperbolic geometry to prove the skein relations. • Our proof is purely combinatorial . The key ingredient to our proof is Theorem 12 where we show the bijection between the perfect matchings. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 15 / 18

  86. Band and snake graph calculus Skein Relations I. Canakci, R. Schiffler Motivation As a corollary we obtain a new proof of the skein relations [MW]. Abstract Snake Graphs Corollary (CS) Relation to Cluster Algebras Let γ 1 and γ 2 be two arcs which cross and let ( γ 3 , γ 4 ) and ( γ 5 , γ 6 ) be Band Graphs and the two pairs of arcs obtained by smoothing the crossing. Then Future Directions x γ 1 x γ 2 = x γ 3 x γ 4 + y ( ˜ G ) x γ 5 x γ 6 where ˜ G is the closure of the overlap G . Remark • Note that Musiker and Williams in [MW] use hyperbolic geometry to prove the skein relations. • Our proof is purely combinatorial . The key ingredient to our proof is Theorem 12 where we show the bijection between the perfect matchings. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 15 / 18

  87. Band and snake graph calculus Skein Relations I. Canakci, R. Schiffler Motivation As a corollary we obtain a new proof of the skein relations [MW]. Abstract Snake Graphs Corollary (CS) Relation to Cluster Algebras Let γ 1 and γ 2 be two arcs which cross and let ( γ 3 , γ 4 ) and ( γ 5 , γ 6 ) be Band Graphs and the two pairs of arcs obtained by smoothing the crossing. Then Future Directions x γ 1 x γ 2 = x γ 3 x γ 4 + y ( ˜ G ) x γ 5 x γ 6 where ˜ G is the closure of the overlap G . Remark • Note that Musiker and Williams in [MW] use hyperbolic geometry to prove the skein relations. • Our proof is purely combinatorial . The key ingredient to our proof is Theorem 12 where we show the bijection between the perfect matchings. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 15 / 18

  88. Band and snake graph calculus Skein Relations I. Canakci, R. Schiffler Motivation As a corollary we obtain a new proof of the skein relations [MW]. Abstract Snake Graphs Corollary (CS) Relation to Cluster Algebras Let γ 1 and γ 2 be two arcs which cross and let ( γ 3 , γ 4 ) and ( γ 5 , γ 6 ) be Band Graphs and the two pairs of arcs obtained by smoothing the crossing. Then Future Directions x γ 1 x γ 2 = x γ 3 x γ 4 + y ( ˜ G ) x γ 5 x γ 6 where ˜ G is the closure of the overlap G . Remark • Note that Musiker and Williams in [MW] use hyperbolic geometry to prove the skein relations. • Our proof is purely combinatorial . The key ingredient to our proof is Theorem 12 where we show the bijection between the perfect matchings. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 15 / 18

  89. Band and snake graph calculus Band Graphs I. Canakci, R. Schiffler • I am currently working on extending our combinatorial formulas Motivation to band graphs associated to closed loops in a surface, see Abstract Snake Graphs [MSW2] . Relation to Cluster Algebras • Closed loops appear naturally in the process of smoothing Band Graphs and crossings. Consider the following example. Future Directions Example In this example we resolve two crossings of the following arcs. • − → + − → + + + • • • Question: Is this construction straightforward? Answer: No! The difficulty here is to show the ’skein relations’ for self-crossing arcs. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 16 / 18

  90. Band and snake graph calculus Band Graphs I. Canakci, R. Schiffler • I am currently working on extending our combinatorial formulas Motivation to band graphs associated to closed loops in a surface, see Abstract Snake Graphs [MSW2] . Relation to Cluster Algebras • Closed loops appear naturally in the process of smoothing Band Graphs and crossings. Consider the following example. Future Directions Example In this example we resolve two crossings of the following arcs. • − → + − → + + + • • • Question: Is this construction straightforward? Answer: No! The difficulty here is to show the ’skein relations’ for self-crossing arcs. I. Canakci, R. Schiffler (U. Conn.) Band and snake graph calculus Auslander Distinguished Lectures 2013 16 / 18

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