On surface cluster algebras: Band and snake Abstract Snake Graphs - - PowerPoint PPT Presentation

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On surface cluster algebras: Band and snake Abstract Snake Graphs - - PowerPoint PPT Presentation

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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

slide-1
SLIDE 1

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

On surface cluster algebras: Band and snake graph calculus

Ilke Canakci1 Ralf Schiffler1

1Department of Mathematics

University of Connecticut

Maurice Auslander Distinguished Lectures and International Conference April 18 - 23, 2013

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 1 / 18

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SLIDE 2

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Background

  • Cluster algebras, introduced by Fomin and Zelevinsky in [FZ1]

form a class of combinatorially defined commutative algebras, and the set of generators of a cluster algebra, cluster variables, is obtained by an iterative process.

  • A surface cluster algebra A(S, M) is associated to a surface S

with boundary that has finitely many marked points.

  • 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

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SLIDE 3

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Background

  • Cluster algebras, introduced by Fomin and Zelevinsky in [FZ1]

form a class of combinatorially defined commutative algebras, and the set of generators of a cluster algebra, cluster variables, is obtained by an iterative process.

  • A surface cluster algebra A(S, M) is associated to a surface S

with boundary that has finitely many marked points.

  • 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

slide-4
SLIDE 4

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Background

  • Cluster algebras, introduced by Fomin and Zelevinsky in [FZ1]

form a class of combinatorially defined commutative algebras, and the set of generators of a cluster algebra, cluster variables, is obtained by an iterative process.

  • A surface cluster algebra A(S, M) is associated to a surface S

with boundary that has finitely many marked points.

1 2 3

  • 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

slide-5
SLIDE 5

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Background

  • Cluster algebras, introduced by Fomin and Zelevinsky in [FZ1]

form a class of combinatorially defined commutative algebras, and the set of generators of a cluster algebra, cluster variables, is obtained by an iterative process.

  • A surface cluster algebra A(S, M) is associated to a surface S

with boundary that has finitely many marked points.

1 2 3

  • 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

slide-6
SLIDE 6

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Background

  • Cluster algebras, introduced by Fomin and Zelevinsky in [FZ1]

form a class of combinatorially defined commutative algebras, and the set of generators of a cluster algebra, cluster variables, is obtained by an iterative process.

  • A surface cluster algebra A(S, M) is associated to a surface S

with boundary that has finitely many marked points.

1 2 3

γ1

  • 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

slide-7
SLIDE 7

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Background

  • Cluster algebras, introduced by Fomin and Zelevinsky in [FZ1]

form a class of combinatorially defined commutative algebras, and the set of generators of a cluster algebra, cluster variables, is obtained by an iterative process.

  • A surface cluster algebra A(S, M) is associated to a surface S

with boundary that has finitely many marked points.

1 2 3

γ1 γ2

  • 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

slide-8
SLIDE 8

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Background

  • Cluster algebras, introduced by Fomin and Zelevinsky in [FZ1]

form a class of combinatorially defined commutative algebras, and the set of generators of a cluster algebra, cluster variables, is obtained by an iterative process.

  • A surface cluster algebra A(S, M) is associated to a surface S

with boundary that has finitely many marked points.

1 2 3

γ1 γ2

  • 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

slide-9
SLIDE 9

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Background

  • Cluster algebras, introduced by Fomin and Zelevinsky in [FZ1]

form a class of combinatorially defined commutative algebras, and the set of generators of a cluster algebra, cluster variables, is obtained by an iterative process.

  • A surface cluster algebra A(S, M) is associated to a surface S

with boundary that has finitely many marked points.

1 2 3

γ1 γ2 γ3 γ4 γ5 γ6

  • 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

slide-10
SLIDE 10

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Background

  • Cluster algebras, introduced by Fomin and Zelevinsky in [FZ1]

form a class of combinatorially defined commutative algebras, and the set of generators of a cluster algebra, cluster variables, is obtained by an iterative process.

  • A surface cluster algebra A(S, M) is associated to a surface S

with boundary that has finitely many marked points.

1 2 3

γ1 γ2 γ3 γ4 γ5 γ6

xγ1xγ2 = ∗xγ3xγ4 + ∗xγ5xγ6

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

slide-11
SLIDE 11

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Background

  • Cluster algebras, introduced by Fomin and Zelevinsky in [FZ1]

form a class of combinatorially defined commutative algebras, and the set of generators of a cluster algebra, cluster variables, is obtained by an iterative process.

  • A surface cluster algebra A(S, M) is associated to a surface S

with boundary that has finitely many marked points.

1 2 3

γ1 γ2 γ3 γ4 γ5 γ6

xγ1xγ2 = ∗xγ3xγ4 + ∗xγ5xγ6

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

slide-12
SLIDE 12

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Motivation

Let A(S, M) cluster algebra associated to a surface (S, M). 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

slide-13
SLIDE 13

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Motivation

Let A(S, M) cluster algebra associated to a surface (S, M). We have the following situation: cluster variable ← →

[FST]

arc

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

slide-14
SLIDE 14

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Motivation

Let A(S, M) cluster algebra associated to a surface (S, M). We have the following situation: cluster variable ← →

[FST]

arc − →

[MSW]

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

slide-15
SLIDE 15

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Motivation

Let A(S, M) cluster algebra associated to a surface (S, M). We have the following situation: cluster variable ← →

[FST]

arc − →

[MSW]

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

slide-16
SLIDE 16

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Motivation

Let A(S, M) cluster algebra associated to a surface (S, M). We have the following situation: cluster variable ← →

[FST]

arc − →

[MSW]

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

slide-17
SLIDE 17

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Motivation

Let A(S, M) cluster algebra associated to a surface (S, M). We have the following situation: cluster variable ← →

[FST]

arc − →

[MSW]

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

slide-18
SLIDE 18

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Motivation

Let A(S, M) cluster algebra associated to a surface (S, M). We have the following situation: cluster variable ← →

[FST]

arc − →

[MSW]

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

slide-19
SLIDE 19

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Abstract Snake Graphs

Definition

A snake graph G is a connected graph in R2 consisting of a finite sequence of tiles G1, G2, . . . , Gd with d ≥ 1, such that for each i = 1, . . . , d − 1 (i) Gi and Gi+1 share exactly one edge ei and this edge is either the north edge of Gi and the south edge of Gi+1 or the east edge of Gi and the west edge of Gi+1. (ii) Gi and Gj have no edge in common whenever |i − j| ≥ 2. (ii) Gi and Gj 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

slide-20
SLIDE 20

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Abstract Snake Graphs

Definition

A snake graph G is a connected graph in R2 consisting of a finite sequence of tiles G1, G2, . . . , Gd with d ≥ 1, such that for each i = 1, . . . , d − 1 (i) Gi and Gi+1 share exactly one edge ei and this edge is either the north edge of Gi and the south edge of Gi+1 or the east edge of Gi and the west edge of Gi+1. (ii) Gi and Gj have no edge in common whenever |i − j| ≥ 2. (ii) Gi and Gj 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

slide-21
SLIDE 21

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Abstract Snake Graphs

Definition

A snake graph G is a connected graph in R2 consisting of a finite sequence of tiles G1, G2, . . . , Gd with d ≥ 1, such that for each i = 1, . . . , d − 1 (i) Gi and Gi+1 share exactly one edge ei and this edge is either the north edge of Gi and the south edge of Gi+1 or the east edge of Gi and the west edge of Gi+1. (ii) Gi and Gj have no edge in common whenever |i − j| ≥ 2. (ii) Gi and Gj 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

slide-22
SLIDE 22

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Example

G G1

Notation

  • G = (G1, G2, . . . , Gd)
  • G[i, i + t] = (Gi, Gi+1, . . . , Gi+t)
  • We denote by ei the interior edge between the tiles Gi and Gi+1.
  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 5 / 18

slide-23
SLIDE 23

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Example

G G1 G2

Notation

  • G = (G1, G2, . . . , Gd)
  • G[i, i + t] = (Gi, Gi+1, . . . , Gi+t)
  • We denote by ei the interior edge between the tiles Gi and Gi+1.
  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 5 / 18

slide-24
SLIDE 24

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Example

G G1 G2

Notation

  • G = (G1, G2, . . . , Gd)
  • G[i, i + t] = (Gi, Gi+1, . . . , Gi+t)
  • We denote by ei the interior edge between the tiles Gi and Gi+1.
  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 5 / 18

slide-25
SLIDE 25

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Example

G G1 G2

Notation

  • G = (G1, G2, . . . , Gd)
  • G[i, i + t] = (Gi, Gi+1, . . . , Gi+t)
  • We denote by ei the interior edge between the tiles Gi and Gi+1.
  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 5 / 18

slide-26
SLIDE 26

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Example

G G1 G2

Notation

  • G = (G1, G2, . . . , Gd)
  • G[i, i + t] = (Gi, Gi+1, . . . , Gi+t)
  • We denote by ei the interior edge between the tiles Gi and Gi+1.
  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 5 / 18

slide-27
SLIDE 27

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Local Overlaps

Definition

We say two snake graphs G1 and G2 have a local overlap G if G is a maximal subgraph contained in both G1 and G2. Notation: G ∼ = G1[s, · · · , t] ∼ = G2[s′, · · · , t′].

Example

G1 G2

Therefore G is a local overlap of G1 and G2.

  • 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

slide-28
SLIDE 28

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Local Overlaps

Definition

We say two snake graphs G1 and G2 have a local overlap G if G is a maximal subgraph contained in both G1 and G2. Notation: G ∼ = G1[s, · · · , t] ∼ = G2[s′, · · · , t′].

Example

G1 G2

Therefore G is a local overlap of G1 and G2.

  • 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

slide-29
SLIDE 29

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Local Overlaps

Definition

We say two snake graphs G1 and G2 have a local overlap G if G is a maximal subgraph contained in both G1 and G2. Notation: G ∼ = G1[s, · · · , t] ∼ = G2[s′, · · · , t′].

Example

G1 G2

Therefore G is a local overlap of G1 and G2.

  • 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

slide-30
SLIDE 30

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Local Overlaps

Definition

We say two snake graphs G1 and G2 have a local overlap G if G is a maximal subgraph contained in both G1 and G2. Notation: G ∼ = G1[s, · · · , t] ∼ = G2[s′, · · · , t′].

Example

G1 G2

Therefore G is a local overlap of G1 and G2.

  • 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

slide-31
SLIDE 31

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Local Overlaps

Definition

We say two snake graphs G1 and G2 have a local overlap G if G is a maximal subgraph contained in both G1 and G2. Notation: G ∼ = G1[s, · · · , t] ∼ = G2[s′, · · · , t′].

Example

G1 G2

Therefore G is a local overlap of G1 and G2.

  • 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

slide-32
SLIDE 32

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Local Overlaps

Definition

We say two snake graphs G1 and G2 have a local overlap G if G is a maximal subgraph contained in both G1 and G2. Notation: G ∼ = G1[s, · · · , t] ∼ = G2[s′, · · · , t′].

Example

G1 G2 G

Therefore G is a local overlap of G1 and G2.

  • 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

slide-33
SLIDE 33

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Local Overlaps

Definition

We say two snake graphs G1 and G2 have a local overlap G if G is a maximal subgraph contained in both G1 and G2. Notation: G ∼ = G1[s, · · · , t] ∼ = G2[s′, · · · , t′].

Example

G1 G2 G

Therefore G is a local overlap of G1 and G2.

  • 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

slide-34
SLIDE 34

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Local Overlaps

Definition

We say two snake graphs G1 and G2 have a local overlap G if G is a maximal subgraph contained in both G1 and G2. Notation: G ∼ = G1[s, · · · , t] ∼ = G2[s′, · · · , t′].

Example

G1 G2 G

Therefore G is a local overlap of G1 and G2.

  • 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

slide-35
SLIDE 35

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Sign Function

Definition

A sign function f on a snake graph G is a map f from the set of 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 G1 and G2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

slide-36
SLIDE 36

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Sign Function

Definition

A sign function f on a snake graph G is a map f from the set of 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 G1 and G2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

slide-37
SLIDE 37

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Sign Function

Definition

A sign function f on a snake graph G is a map f from the set of 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 G1 and G2

− + − +

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

slide-38
SLIDE 38

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Sign Function

Definition

A sign function f on a snake graph G is a map f from the set of 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 G1 and G2

− + − + G1

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

slide-39
SLIDE 39

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Sign Function

Definition

A sign function f on a snake graph G is a map f from the set of 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 G1 and G2

− + − + G1

+

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

slide-40
SLIDE 40

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Sign Function

Definition

A sign function f on a snake graph G is a map f from the set of 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 G1 and G2

− + − + G1

+ +

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

slide-41
SLIDE 41

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Sign Function

Definition

A sign function f on a snake graph G is a map f from the set of 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 G1 and G2

− + − + G1

+ + −

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

slide-42
SLIDE 42

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Sign Function

Definition

A sign function f on a snake graph G is a map f from the set of 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 G1 and G2

− + − + G1

+ + − + + −− −− −−+ + − − − +

−

+ + ++ − +

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

slide-43
SLIDE 43

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Sign Function

Definition

A sign function f on a snake graph G is a map f from the set of 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 G1 and G2

G1

+ + − + + −− −− −−+ + − − − +

−

+ + ++ − +

G2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

slide-44
SLIDE 44

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Sign Function

Definition

A sign function f on a snake graph G is a map f from the set of 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 G1 and G2

G1

+ + − + + −− −− −−+ + − − − +

−

+ + ++ − + +

G2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

slide-45
SLIDE 45

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Sign Function

Definition

A sign function f on a snake graph G is a map f from the set of 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 G1 and G2

G1

+ + − + + −− −− −−+ + − − − +

−

+ + ++ − + + − − − ++− − − − + − − − +

G2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 7 / 18

slide-46
SLIDE 46

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Crossing

Definition

We say that G1 and G2 cross in a local overlap G if one of the following conditions hold.

  • f1(es−1) = −f1(et) if s > 1, t < d
  • f1(es−1) = f2(e′

t′) if s > 1, t < d, s′ = 1, t′ < d′

Example

G1 and G2 cross at the overlap G.

G1 G2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 8 / 18

slide-47
SLIDE 47

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Crossing

Definition

We say that G1 and G2 cross in a local overlap G if one of the following conditions hold.

  • f1(es−1) = −f1(et) if s > 1, t < d
  • f1(es−1) = f2(e′

t′) if s > 1, t < d, s′ = 1, t′ < d′

Example

G1 and G2 cross at the overlap G.

G1 G2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 8 / 18

slide-48
SLIDE 48

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Crossing

Definition

We say that G1 and G2 cross in a local overlap G if one of the following conditions hold.

  • f1(es−1) = −f1(et) if s > 1, t < d
  • f1(es−1) = f2(e′

t′) if s > 1, t < d, s′ = 1, t′ < d′

Example

G1 and G2 cross at the overlap G.

G1 G2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 8 / 18

slide-49
SLIDE 49

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Crossing

Definition

We say that G1 and G2 cross in a local overlap G if one of the following conditions hold.

  • f1(es−1) = −f1(et) if s > 1, t < d
  • f1(es−1) = f2(e′

t′) if s > 1, t < d, s′ = 1, t′ < d′

Example

G1 and G2 cross at the overlap G.

G1 G2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 8 / 18

slide-50
SLIDE 50

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Crossing

Definition

We say that G1 and G2 cross in a local overlap G if one of the following conditions hold.

  • f1(es−1) = −f1(et) if s > 1, t < d
  • f1(es−1) = f2(e′

t′) if s > 1, t < d, s′ = 1, t′ < d′

Example

G1 and G2 cross at the overlap G.

G1

+

G2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 8 / 18

slide-51
SLIDE 51

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Crossing

Definition

We say that G1 and G2 cross in a local overlap G if one of the following conditions hold.

  • f1(es−1) = −f1(et) if s > 1, t < d
  • f1(es−1) = f2(e′

t′) if s > 1, t < d, s′ = 1, t′ < d′

Example

G1 and G2 cross at the overlap G.

G1

+ −

G2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 8 / 18

slide-52
SLIDE 52

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Crossing

Definition

We say that G1 and G2 cross in a local overlap G if one of the following conditions hold.

  • f1(es−1) = −f1(et) if s > 1, t < d
  • f1(es−1) = f2(e′

t′) if s > 1, t < d, s′ = 1, t′ < d′

Example

G1 and G2 cross at the overlap G.

G1

+ −

G2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 8 / 18

slide-53
SLIDE 53

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Crossing

Definition

We say that G1 and G2 cross in a local overlap G if one of the following conditions hold.

  • f1(es−1) = −f1(et) if s > 1, t < d
  • f1(es−1) = f2(e′

t′) if s > 1, t < d, s′ = 1, t′ < d′

Example

G1 and G2 cross at the overlap G.

G1

+ −

G2 +

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 8 / 18

slide-54
SLIDE 54

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Example: Resolution Res G(G1, G2)

G1 G2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 9 / 18

slide-55
SLIDE 55

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Example: Resolution Res G(G1, G2)

G1 G2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 9 / 18

slide-56
SLIDE 56

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Example: Resolution Res G(G1, G2)

G1 G2 G3

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 9 / 18

slide-57
SLIDE 57

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Example: Resolution Res G(G1, G2)

G1 G2 G3

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 9 / 18

slide-58
SLIDE 58

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Example: Resolution Res G(G1, G2)

G1 G2 G3 G4

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 9 / 18

slide-59
SLIDE 59

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Example: Resolution Res G(G1, G2)

G1 G2 G3 G4

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 9 / 18

slide-60
SLIDE 60

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Example: Resolution Res G(G1, G2)

+ G1 G2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 9 / 18

slide-61
SLIDE 61

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Example: Resolution Res G(G1, G2)

− − − − − −

+ G1 G2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 9 / 18

slide-62
SLIDE 62

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Example: Resolution Res G(G1, G2)

+

− − − − − −

+ G1 G2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 9 / 18

slide-63
SLIDE 63

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Example: Resolution Res G(G1, G2)

+

− − − − − −

+ G1 G2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 9 / 18

slide-64
SLIDE 64

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Example: Resolution (Continued)

G1 G2 G3 G4 G5

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 10 / 18

slide-65
SLIDE 65

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Example: Resolution (Continued)

G1 G2 G3 G4 G5 G6

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 10 / 18

slide-66
SLIDE 66

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Example: Resolution (Continued)

G1 G2 G3 G4 G5 G6

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 10 / 18

slide-67
SLIDE 67

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Resolution: Definition

Assumption: We will assume that s > 1, t < d, s′ = 1 and t′ < d′. For all other cases, see [CS].

We define four connected subgraphs as follows.

  • G3 = G1[1, t] ∪ G2[t′ + 1, d′],
  • G4 = G2[1, t′] ∪ G1[t + 1, d],
  • G5 = G1[1, k] where k < s − 1 is the largest integer such that the sign
  • n the interior edge between tiles k and k + 1 is the same as the sign
  • n the interior edge of tiles s − 1 and s,
  • G6 = G2[d′, t′ + 1] ∪ G1[t + 1, d] where the two subgraphs are glued

along the south Gt+1 and the north of G ′

t′+1 if Gt+1 is north of Gt in

G1.

Definition

The resolution of the crossing of G1 and G2 in G is defined to be (G3 ⊔ G4, G5 ⊔ G6) and is denoted by Res G(G1, G2).

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 11 / 18

slide-68
SLIDE 68

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Resolution: Definition

Assumption: We will assume that s > 1, t < d, s′ = 1 and t′ < d′. For all other cases, see [CS].

We define four connected subgraphs as follows.

  • G3 = G1[1, t] ∪ G2[t′ + 1, d′],
  • G4 = G2[1, t′] ∪ G1[t + 1, d],
  • G5 = G1[1, k] where k < s − 1 is the largest integer such that the sign
  • n the interior edge between tiles k and k + 1 is the same as the sign
  • n the interior edge of tiles s − 1 and s,
  • G6 = G2[d′, t′ + 1] ∪ G1[t + 1, d] where the two subgraphs are glued

along the south Gt+1 and the north of G ′

t′+1 if Gt+1 is north of Gt in

G1.

Definition

The resolution of the crossing of G1 and G2 in G is defined to be (G3 ⊔ G4, G5 ⊔ G6) and is denoted by Res G(G1, G2).

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 11 / 18

slide-69
SLIDE 69

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Resolution: Definition

Assumption: We will assume that s > 1, t < d, s′ = 1 and t′ < d′. For all other cases, see [CS].

We define four connected subgraphs as follows.

  • G3 = G1[1, t] ∪ G2[t′ + 1, d′],
  • G4 = G2[1, t′] ∪ G1[t + 1, d],
  • G5 = G1[1, k] where k < s − 1 is the largest integer such that the sign
  • n the interior edge between tiles k and k + 1 is the same as the sign
  • n the interior edge of tiles s − 1 and s,
  • G6 = G2[d′, t′ + 1] ∪ G1[t + 1, d] where the two subgraphs are glued

along the south Gt+1 and the north of G ′

t′+1 if Gt+1 is north of Gt in

G1.

Definition

The resolution of the crossing of G1 and G2 in G is defined to be (G3 ⊔ G4, G5 ⊔ G6) and is denoted by Res G(G1, G2).

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 11 / 18

slide-70
SLIDE 70

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Resolution: Definition

Assumption: We will assume that s > 1, t < d, s′ = 1 and t′ < d′. For all other cases, see [CS].

We define four connected subgraphs as follows.

  • G3 = G1[1, t] ∪ G2[t′ + 1, d′],
  • G4 = G2[1, t′] ∪ G1[t + 1, d],
  • G5 = G1[1, k] where k < s − 1 is the largest integer such that the sign
  • n the interior edge between tiles k and k + 1 is the same as the sign
  • n the interior edge of tiles s − 1 and s,
  • G6 = G2[d′, t′ + 1] ∪ G1[t + 1, d] where the two subgraphs are glued

along the south Gt+1 and the north of G ′

t′+1 if Gt+1 is north of Gt in

G1.

Definition

The resolution of the crossing of G1 and G2 in G is defined to be (G3 ⊔ G4, G5 ⊔ G6) and is denoted by Res G(G1, G2).

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 11 / 18

slide-71
SLIDE 71

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Resolution: Definition

Assumption: We will assume that s > 1, t < d, s′ = 1 and t′ < d′. For all other cases, see [CS].

We define four connected subgraphs as follows.

  • G3 = G1[1, t] ∪ G2[t′ + 1, d′],
  • G4 = G2[1, t′] ∪ G1[t + 1, d],
  • G5 = G1[1, k] where k < s − 1 is the largest integer such that the sign
  • n the interior edge between tiles k and k + 1 is the same as the sign
  • n the interior edge of tiles s − 1 and s,
  • G6 = G2[d′, t′ + 1] ∪ G1[t + 1, d] where the two subgraphs are glued

along the south Gt+1 and the north of G ′

t′+1 if Gt+1 is north of Gt in

G1.

Definition

The resolution of the crossing of G1 and G2 in G is defined to be (G3 ⊔ G4, G5 ⊔ G6) and is denoted by Res G(G1, G2).

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 11 / 18

slide-72
SLIDE 72

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Bijection of Perfect Matchings

Definition

A perfect matching P of a graph G is a subset of the set of edges

  • f G such that each vertex of G is incident to exactly one edge in P.
  • Let Match (G) denote the set of all perfect matchings of the

graph G and Match (Res G(G1, G2)) = Match (G3 ⊔ G4) ∪ Match (G5 ⊔ G6).

Theorem (CS)

Let G1, G2 be two snake graphs. Then there is a bijection Match (G1 ⊔ G2) − → Match (Res G(G1, G2))

  • 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

slide-73
SLIDE 73

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Bijection of Perfect Matchings

Definition

A perfect matching P of a graph G is a subset of the set of edges

  • f G such that each vertex of G is incident to exactly one edge in P.
  • Let Match (G) denote the set of all perfect matchings of the

graph G and Match (Res G(G1, G2)) = Match (G3 ⊔ G4) ∪ Match (G5 ⊔ G6).

Theorem (CS)

Let G1, G2 be two snake graphs. Then there is a bijection Match (G1 ⊔ G2) − → Match (Res G(G1, G2))

  • 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

slide-74
SLIDE 74

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Bijection of Perfect Matchings

Definition

A perfect matching P of a graph G is a subset of the set of edges

  • f G such that each vertex of G is incident to exactly one edge in P.
  • Let Match (G) denote the set of all perfect matchings of the

graph G and Match (Res G(G1, G2)) = Match (G3 ⊔ G4) ∪ Match (G5 ⊔ G6).

Theorem (CS)

Let G1, G2 be two snake graphs. Then there is a bijection Match (G1 ⊔ G2) − → Match (Res G(G1, G2))

  • 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

slide-75
SLIDE 75

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Bijection of Perfect Matchings

Definition

A perfect matching P of a graph G is a subset of the set of edges

  • f G such that each vertex of G is incident to exactly one edge in P.
  • Let Match (G) denote the set of all perfect matchings of the

graph G and Match (Res G(G1, G2)) = Match (G3 ⊔ G4) ∪ Match (G5 ⊔ G6).

Theorem (CS)

Let G1, G2 be two snake graphs. Then there is a bijection Match (G1 ⊔ G2) − → Match (Res G(G1, G2))

  • 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

slide-76
SLIDE 76

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Bijection of Perfect Matchings

Definition

A perfect matching P of a graph G is a subset of the set of edges

  • f G such that each vertex of G is incident to exactly one edge in P.
  • Let Match (G) denote the set of all perfect matchings of the

graph G and Match (Res G(G1, G2)) = Match (G3 ⊔ G4) ∪ Match (G5 ⊔ G6).

Theorem (CS)

Let G1, G2 be two snake graphs. Then there is a bijection Match (G1 ⊔ G2) − → Match (Res G(G1, G2))

  • 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

slide-77
SLIDE 77

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Bijection of Perfect Matchings

Definition

A perfect matching P of a graph G is a subset of the set of edges

  • f G such that each vertex of G is incident to exactly one edge in P.
  • Let Match (G) denote the set of all perfect matchings of the

graph G and Match (Res G(G1, G2)) = Match (G3 ⊔ G4) ∪ Match (G5 ⊔ G6).

Theorem (CS)

Let G1, G2 be two snake graphs. Then there is a bijection Match (G1 ⊔ G2) − → Match (Res G(G1, G2))

  • 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

slide-78
SLIDE 78

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Surface Example

1 2 3 4 5 6 7 8 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

slide-79
SLIDE 79

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Surface Example

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

γ1

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

slide-80
SLIDE 80

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Surface Example

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

γ1 γ2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

slide-81
SLIDE 81

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Surface Example

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

γ1 γ2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

G1

13 14 15 16 17 18 26 27 28 29 30 31 32 33 34 35

G2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

slide-82
SLIDE 82

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Surface Example

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

γ1 γ2

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

slide-83
SLIDE 83

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Surface Example

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

γ1 γ2 γ3

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

slide-84
SLIDE 84

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Surface Example

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

γ1 γ2 γ3 γ4

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

slide-85
SLIDE 85

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Surface Example

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

γ1 γ2 γ3 γ4 γ5

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

slide-86
SLIDE 86

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Surface Example

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

γ1 γ2 γ3 γ4 γ5 γ6

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

slide-87
SLIDE 87

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Surface Example

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 26 27 28 29 30 31 32 33 34 35

G3

13 14 15 16 17 18 19 20 21 22 23 24 25

G4

1 2 3 4 5

G5

26 27 28 29 30 31 32 33 34 35 19 20 21 22 23 24 25

G6

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

slide-88
SLIDE 88

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Surface Example

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

γ1 γ2 γ3 γ4 γ5 γ6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

G1

13 14 15 16 17 18 26 27 28 29 30 31 32 33 34 35

G2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 26 27 28 29 30 31 32 33 34 35

G3

13 14 15 16 17 18 19 20 21 22 23 24 25

G4

1 2 3 4 5

G5

26 27 28 29 30 31 32 33 34 35 19 20 21 22 23 24 25

G6

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

slide-89
SLIDE 89

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Surface Example

G1 G2 G3 G4 G5 G6

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 13 / 18

slide-90
SLIDE 90

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Relation to Cluster Algebras

Let γ1 and γ2 be two arcs and G1 and G2 their corresponding snake graphs.

Theorem (CS)

γ1 and γ2 cross if and only if G1 and G2 cross.

Theorem (CS)

If γ1 and γ2 cross, then the snake graphs of the four arcs obtained by smoothing the crossing are given by the resolution Res G(G1, G2)

  • f the crossing of the snake graphs G1 and G2 at the overlap G.

Remark

We do not assume that γ1 and γ2 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

slide-91
SLIDE 91

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Relation to Cluster Algebras

Let γ1 and γ2 be two arcs and G1 and G2 their corresponding snake graphs.

Theorem (CS)

γ1 and γ2 cross if and only if G1 and G2 cross.

Theorem (CS)

If γ1 and γ2 cross, then the snake graphs of the four arcs obtained by smoothing the crossing are given by the resolution Res G(G1, G2)

  • f the crossing of the snake graphs G1 and G2 at the overlap G.

Remark

We do not assume that γ1 and γ2 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

slide-92
SLIDE 92

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Relation to Cluster Algebras

Let γ1 and γ2 be two arcs and G1 and G2 their corresponding snake graphs.

Theorem (CS)

γ1 and γ2 cross if and only if G1 and G2 cross.

Theorem (CS)

If γ1 and γ2 cross, then the snake graphs of the four arcs obtained by smoothing the crossing are given by the resolution Res G(G1, G2)

  • f the crossing of the snake graphs G1 and G2 at the overlap G.

Remark

We do not assume that γ1 and γ2 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

slide-93
SLIDE 93

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Relation to Cluster Algebras

Let γ1 and γ2 be two arcs and G1 and G2 their corresponding snake graphs.

Theorem (CS)

γ1 and γ2 cross if and only if G1 and G2 cross.

Theorem (CS)

If γ1 and γ2 cross, then the snake graphs of the four arcs obtained by smoothing the crossing are given by the resolution Res G(G1, G2)

  • f the crossing of the snake graphs G1 and G2 at the overlap G.

Remark

We do not assume that γ1 and γ2 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

slide-94
SLIDE 94

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Skein Relations

As a corollary we obtain a new proof of the skein relations [MW].

Corollary (CS)

Let γ1 and γ2 be two arcs which cross and let (γ3, γ4) and (γ5, γ6) be the two pairs of arcs obtained by smoothing the crossing. Then xγ1xγ2 = xγ3xγ4 + y( ˜ G)xγ5xγ6 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

slide-95
SLIDE 95

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Skein Relations

As a corollary we obtain a new proof of the skein relations [MW].

Corollary (CS)

Let γ1 and γ2 be two arcs which cross and let (γ3, γ4) and (γ5, γ6) be the two pairs of arcs obtained by smoothing the crossing. Then xγ1xγ2 = xγ3xγ4 + y( ˜ G)xγ5xγ6 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

slide-96
SLIDE 96

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Skein Relations

As a corollary we obtain a new proof of the skein relations [MW].

Corollary (CS)

Let γ1 and γ2 be two arcs which cross and let (γ3, γ4) and (γ5, γ6) be the two pairs of arcs obtained by smoothing the crossing. Then xγ1xγ2 = xγ3xγ4 + y( ˜ G)xγ5xγ6 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

slide-97
SLIDE 97

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Skein Relations

As a corollary we obtain a new proof of the skein relations [MW].

Corollary (CS)

Let γ1 and γ2 be two arcs which cross and let (γ3, γ4) and (γ5, γ6) be the two pairs of arcs obtained by smoothing the crossing. Then xγ1xγ2 = xγ3xγ4 + y( ˜ G)xγ5xγ6 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

slide-98
SLIDE 98

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Skein Relations

As a corollary we obtain a new proof of the skein relations [MW].

Corollary (CS)

Let γ1 and γ2 be two arcs which cross and let (γ3, γ4) and (γ5, γ6) be the two pairs of arcs obtained by smoothing the crossing. Then xγ1xγ2 = xγ3xγ4 + y( ˜ G)xγ5xγ6 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

slide-99
SLIDE 99

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Band Graphs

  • I am currently working on extending our combinatorial formulas

to band graphs associated to closed loops in a surface, see

[MSW2].

  • Closed loops appear naturally in the process of smoothing
  • crossings. Consider the following example.

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

slide-100
SLIDE 100

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Band Graphs

  • I am currently working on extending our combinatorial formulas

to band graphs associated to closed loops in a surface, see

[MSW2].

  • Closed loops appear naturally in the process of smoothing
  • crossings. Consider the following example.

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

slide-101
SLIDE 101

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Band Graphs

  • I am currently working on extending our combinatorial formulas

to band graphs associated to closed loops in a surface, see

[MSW2].

  • Closed loops appear naturally in the process of smoothing
  • crossings. Consider the following example.

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

slide-102
SLIDE 102

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Band Graphs

  • I am currently working on extending our combinatorial formulas

to band graphs associated to closed loops in a surface, see

[MSW2].

  • Closed loops appear naturally in the process of smoothing
  • crossings. Consider the following example.

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

slide-103
SLIDE 103

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Band Graphs

  • I am currently working on extending our combinatorial formulas

to band graphs associated to closed loops in a surface, see

[MSW2].

  • Closed loops appear naturally in the process of smoothing
  • crossings. Consider the following example.

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

slide-104
SLIDE 104

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Band Graphs

  • I am currently working on extending our combinatorial formulas

to band graphs associated to closed loops in a surface, see

[MSW2].

  • Closed loops appear naturally in the process of smoothing
  • crossings. Consider the following example.

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

slide-105
SLIDE 105

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Thank you!

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 17 / 18

slide-106
SLIDE 106

Band and snake graph calculus

  • I. Canakci,
  • R. Schiffler

Motivation Abstract Snake Graphs Relation to Cluster Algebras Band Graphs and Future Directions

Bibliography

  • I. Canakci, R. Schiffler Snake graph calculus and cluster algebras from

surfaces, to appear in Journal of Algebra, preprint available at arxiv:1209.4617v1.

  • S. Fomin, M. Shapiro and D. Thurston, Cluster algebras and

triangulated surfaces. Part I: Cluster complexes, Acta Math. 201 (2008), 83-146.

  • S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer.
  • Math. Soc. 15 (2002), 497–529.
  • G. Musiker, R. Schiffler and L. Williams, Positivity for cluster

algebras from surfaces, Adv. Math. 227, (2011), 2241–2308.

  • G. Musiker, R. Schiffler and L. Williams, Bases for cluster algebras

from surfaces, to appear in Compos. Math.

  • G. Musiker and L. Williams, Matrix formulae and skein relations

for cluster algebras from surfaces, preprint, arXiv:1108.3382.

  • I. Canakci, R. Schiffler (U. Conn.)

Band and snake graph calculus Auslander Distinguished Lectures 2013 18 / 18