Snake graphs from orbifolds Elizabeth Kelley (Joint work with - - PowerPoint PPT Presentation

Snake graphs from orbifolds

Elizabeth Kelley (Joint work with Esther Banaian) April 14, 2019

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 1 / 25

Basic Definitions

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 2 / 25

Basic Definitions

Fix a semifield (P, ⊕, ·). Let F be isomorphic to the field of rational functions in n independent variables with coefficients in QP.

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 2 / 25

Basic Definitions

Fix a semifield (P, ⊕, ·). Let F be isomorphic to the field of rational functions in n independent variables with coefficients in QP. Definition: A labeled seed (of geometric type) in F is a triple Σ = (x, y, B) where: x = (x1, . . . , xn) is a free generating set for F; y = (y1, . . . , yn) is an n-tuple with elements in P; and B = (bij) is a skew-symmetrizable n × n integer matrix. We call x the cluster, y the coefficient tuple, and B the exchange matrix

• f the seed (x, y, B).

Cluster variables are related by standard binomial exchange relations.

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 2 / 25

Cluster algebras of surface type

Some cluster algebras can be associated with triangulated surfaces, via the following dictionary: initial cluster seed Σ ↔ ideal triangulation T initial cluster variable xi ↔ initial arc τi ∈ T

• ther cluster variables ↔ other arcs in (S, M)

mutation µk ↔ “flipping” arc τk

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 3 / 25

Cluster algebras of surface type

Some cluster algebras can be associated with triangulated surfaces, via the following dictionary: initial cluster seed Σ ↔ ideal triangulation T initial cluster variable xi ↔ initial arc τi ∈ T

• ther cluster variables ↔ other arcs in (S, M)

mutation µk ↔ “flipping” arc τk

Σ

1 2 3 τ1 τ2 τ3 ↔ ↔

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 3 / 25

Example

Some cluster algebras can be associated with triangulated surfaces, via the following dictionary: initial cluster seed Σ ↔ ideal triangulation T initial cluster variable xi ↔ initial arc τi ∈ T

• ther cluster variables ↔ other arcs in (S, M)

mutation µk ↔ “flipping” arc τk

Σ

µ1

1 2 3 τ1 τ2 τ3 τ1′ ↔ ↔

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 3 / 25

Example

Some cluster algebras can be associated with triangulated surfaces, via the following dictionary: initial cluster seed Σ ↔ ideal triangulation T initial cluster variable xi ↔ initial arc τi ∈ T

• ther cluster variables ↔ other arcs in (S, M)

mutation µk ↔ “flipping” arc τk

Σ

µ1

1 2 3 τ2 τ3 τ1′ ↔ ↔

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 4 / 25

Example

Some cluster algebras can be associated with triangulated surfaces, via the following dictionary: initial cluster seed Σ ↔ ideal triangulation T initial cluster variable xi ↔ initial arc τi ∈ T

• ther cluster variables ↔ other arcs in (S, M)

mutation µk ↔ “flipping” arc τk

Σ

µ1

1 2 3 τ2 τ3 τ1′ τ2′ ↔ ↔

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 5 / 25

Example

Some cluster algebras can be associated with triangulated surfaces, via the following dictionary: initial cluster seed Σ ↔ ideal triangulation T initial cluster variable xi ↔ initial arc τi ∈ T

• ther cluster variables ↔ other arcs in (S, M)

mutation µk ↔ “flipping” arc τk

Σ

µ1

1 2 3 τ3 τ1′ τ2′ ↔ ↔

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 6 / 25

Cluster algebras of surface type

For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs.

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Cluster algebras of surface type

For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs.

τ1 τ2 τ3 f a e b d c a b 2 c 1 1 3 f 2 2 e d 3

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Cluster algebras of surface type

For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs.

τ1 τ2 τ3 γ f a e b d c a b 2 c 1 1 3 f 2 2 e d 3

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Cluster algebras of surface type

For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs.

τ1 τ2 τ3 γ f a e b d c a b 2 c 1 1 3 f 2 2 e d 3

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Cluster algebras of surface type

For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs.

τ1 τ2 τ3 γ f a e b d c b a c τ2 τ1

+

1 3 f 2 2 e d 3

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Cluster algebras of surface type

For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs.

τ1 τ2 τ3 γ f a e b d c a b τ2 c τ1

−

1 3 f 2 2 e d 3

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Cluster algebras of surface type

For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs.

τ1 τ2 τ3 γ f a e b d c a b τ2 c τ1

−

τ1 τ3 f c τ2

+

2 e d 3

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Cluster algebras of surface type

For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs.

τ1 τ2 τ3 γ f a e b d c a b τ2 c τ1

−

τ1 τ3 f τ2

+

2 e d 3

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Cluster algebras of surface type

For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs.

τ1 τ2 τ3 γ f a e b d c a b τ2 c τ1

−

τ1 τ3 f τ2

+

f d e τ2 τ3

+

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Cluster algebras of surface type

For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs.

τ1 τ2 τ3 γ f a e b d c a b τ2 c τ1

−

τ1 τ3 f τ2

+

τ2 e d f τ3

−

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Cluster algebras of surface type

For these algebras, MSW gave an elementary and constructive combinatorial proof of positivity via snake graphs.

τ1 τ2 τ3 γ f a e b d c a b τ2 c τ1

−

τ1 τ3 f τ2

+

τ2 e d τ3

−

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 7 / 25

Theorem: (Musiker-Schiffler-Williams, 2011) Let (S, M) be a bordered surface with triangulation T, A be the corresponding cluster algebra with principal coeficients, and γ be an ordinary arc on S. Then xγ can be written as a Laurent expansion in terms of the initial cluster variables as xγ = 1 cross(T, γ)

• P

x(P)y(P) where P is a perfect matching of GT,γ.

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 8 / 25

Theorem: (Musiker-Schiffler-Williams, 2011) Let (S, M) be a bordered surface with triangulation T, A be the corresponding cluster algebra with principal coeficients, and γ be an ordinary arc on S. Then xγ can be written as a Laurent expansion in terms of the initial cluster variables as xγ = 1 cross(T, γ)

• P

x(P)y(P) where P is a perfect matching of GT,γ. Note: Here, coefficients in the numerator are counting the number of perfect matchings with particular values of x(P) and y(P). Hence, the coefficients must be in Z≥0.

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 8 / 25

xγ = 1 cross(T, γ)

• P

x(P)y(P)

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 9 / 25

xγ = 1 cross(T, γ)

• P

x(P)y(P) cross(T, γ) = the crossing monomial of γ = xi1 · · · xid, for τi1, . . . , τid crossed by γ

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 9 / 25

xγ = 1 cross(T, γ)

• P

x(P)y(P) cross(T, γ) = the crossing monomial of γ = xi1 · · · xid, for τi1, . . . , τid crossed by γ x(P) = (the weight of P) = xi1 · · · xik, where τi1, . . . , τik are the labels of edges of P.

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 9 / 25

xγ = 1 cross(T, γ)

• P

x(P)y(P) cross(T, γ) = the crossing monomial of γ = xi1 · · · xid, for τi1, . . . , τid crossed by γ x(P) = (the weight of P) = xi1 · · · xik, where τi1, . . . , τik are the labels of edges of P. y(P) = (the height of P) = n

k=1 hmk τk , where

hτk =        yτk if τk is not an edge of a self-folded triangle

yr yr(p)

if τk is the radius r to puncture p in a self-folded triangle yr(p) if τk is the loop in a self-folded triangle with radius r to p

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 9 / 25

τ1 τ2 τ3 γ f a e b d c a b τ2 c τ1 τ1 τ3 f τ2 τ2 e d τ3 y2 y1y2x2 y1y2y3x2

2

y2y3x2 x1x3

+ + + +

x1x2x3 xγ =

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 10 / 25

τ1 τ2 τ3 γ f a e b d c a b τ2 c τ1 τ1 τ3 f τ2 τ2 e d τ3 y2 y1y2x2 y1y2y3x2

2

y2y3x2 x1x3

+ + + +

x1x2x3 xγ =

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 10 / 25

τ1 τ2 τ3 γ f a e b d c a b τ2 c τ1 τ1 τ3 f τ2 τ2 e d τ3 y2 y1y2x2 y1y2y3x2

2

y2y3x2 x1x3

+ + + +

x1x2x3 xγ =

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 10 / 25

τ1 τ2 τ3 γ f a e b d c a b τ2 c τ1 τ1 τ3 f τ2 τ2 e d τ3 y2 y1y2x2 y1y2y3x2

2

y2y3x2 x1x3

+ + + +

x1x2x3 xγ =

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 10 / 25

τ1 τ2 τ3 γ f a e b d c a b τ2 c τ1 τ1 τ3 f τ2 τ2 e d τ3 y2 y1y2x2 y1y2y3x2

2

y2y3x2 x1x3

+ + + +

x1x2x3 xγ =

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 10 / 25

τ1 τ2 τ3 γ f a e b d c a b τ2 c τ1 τ1 τ3 f τ2 τ2 e d τ3 y2 y1y2x2 y1y2y3x2

2

y2y3x2 x1x3

+ + + +

x1x2x3 xγ =

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 10 / 25

Cluster Algebras from Orbifolds

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 11 / 25

Cluster Algebras from Orbifolds

Definition: An orbifold is a generalization of a manifold where the local structure is given by quotients of open subsets of Rn under finite group actions. We can think of an orbifold as a surface with isolated singularities at

• rbifold points. Each orbifold point has an associated positive integer
• rder, p.

Definition: An orbifold point of order p has the associated constant λp = 2 cos π p

• Elizabeth Kelley

Snake graphs from orbifolds April 14, 2019 11 / 25

Cluster Algebras from Orbifolds

Intuitively, an orbifold point of order p is “1/pth” of a point. A winding arc with k self-intersections “sees” the orbifold point as a puncture if k < p. an ordinary point if k = p. (Hence, a winding arc with exactly p self-intersections is isotopic to an arc with no self-intersections) When k > p, it behaves like an arc with k mod p self-intersections.

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 12 / 25

Winding for p = 4:

× × ×

b b b b a a a a

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 13 / 25

Winding for p = 4:

× × ×

b b b b a a a a

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 13 / 25

Generalized Cluster Algebras

In ordinary cluster algebras, all exchange polynomials are binomials. In one type of generalized cluster algebra, due to Chekhov and Shapiro, this requirement is relaxed and the exchange polynomials are allowed to have arbitrarily many terms. Cluster algebras from triangulated orbifolds are a particular subclass of these generalized cluster algebras.

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 14 / 25

Results for generalized cluster algebras

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 15 / 25

Results for generalized cluster algebras

Theorem (Chekhov-Shapiro, 2011): Let A = (x, y, B) be an arbitrary generalized cluster algebra. The cluster variables x1, . . . , xn can be expressed in terms of any cluster of A as Laurent polynomials with coefficients in ZP.

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 15 / 25

Results for generalized cluster algebras

Theorem (Chekhov-Shapiro, 2011): Let A = (x, y, B) be an arbitrary generalized cluster algebra. The cluster variables x1, . . . , xn can be expressed in terms of any cluster of A as Laurent polynomials with coefficients in ZP. Theorem (Chekhov-Shapiro, 2011): Positivity holds for generalized cluster algebras from orbifolds.

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 15 / 25

×

1 3 2 a b

b b b a a a

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 16 / 25

×

1 3 2 a b

b b b a a a

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 16 / 25

×

1 3 2 a b

b b b a a a

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 17 / 25

× ×

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 18 / 25

× ×

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 18 / 25

Hence, we want rules for working “downstairs”.

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 19 / 25

Hence, we want rules for working “downstairs”. We already have rules for square tiles due to crossing ordinary arcs.

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 19 / 25

Hence, we want rules for working “downstairs”. We already have rules for square tiles due to crossing ordinary arcs. When crossing pending arcs, there’s a clear labeling for one triangle:

×

1 3 2 a b

b b b a a a

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 19 / 25

Hence, we want rules for working “downstairs”. We already have rules for square tiles due to crossing ordinary arcs. When crossing pending arcs, there’s a clear labeling for one triangle:

×

1 3 2 a b

b b b a a a

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 19 / 25

Hence, we want rules for working “downstairs”. We already have rules for square tiles due to crossing ordinary arcs. When crossing pending arcs, there’s a clear labeling for one triangle:

×

1 3 2 a b

b b b b a a a a

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 19 / 25

Hence, we want rules for working “downstairs”. We already have rules for square tiles due to crossing ordinary arcs. When crossing pending arcs, there’s a clear labeling for one triangle:

×

1 3 2 a b

b b b b b a a a a a

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 19 / 25

Hence, we want rules for working “downstairs”. We already have rules for square tiles due to crossing ordinary arcs. When crossing pending arcs, there’s a clear labeling for one triangle:

×

1 3 2 a b

b b b b b a a a a a

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 19 / 25

Hence, we want rules for working “downstairs”. We already have rules for square tiles due to crossing ordinary arcs. When crossing pending arcs, there’s a clear labeling for one triangle:

τ3 τ2 λpτ2 τ1 τ2

τ3 τ1 τ2 τ2 λpτ2 b b b b b a a a a a

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 19 / 25

An ordinary arc crossing a pending arc produces a tile of the form

τj fτi gτi τk τi

where f , g take values from {1, λp} based on the local configuration:

×

τ2 a τ1

×

τ2 a τ1

τ1 τ1 τ2 τ2 λpτ1 τ1 a τ1 τ2 a τ1 τ1 τ1 τ2 a λpτ1 τ1 τ2 τ2 a τ1 τ1

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 20 / 25

We can assemble longer snake graphs via “gluing” these puzzle pieces.

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 21 / 25

We can assemble longer snake graphs via “gluing” these puzzle pieces. For example:

× ×

τ2 τ1

a

=

• ×

×

τ2 τ1

a

× ×

τ2 τ1

a

τ1 τ1 τ2 τ2 τ2 µτ1 τ1 a τ1 τ2 τ2 λτ2 a τ1 a τ2 τ1 τ1 τ1 τ2 τ2 µτ1 τ1 a τ1 τ2 a τ1 =

• τ1

τ2 τ2 a τ1 τ2 τ1 a τ2 λτ2 τ2

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 21 / 25

Winding arcs with one self-intersection produce the same type of tiles.

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 22 / 25

Winding arcs with one self-intersection produce the same type of tiles.

τ2 a τ1

×

τ2 a τ1

×

τ1 τ1

λpτ1 τ2 a τ1 τ1 a τ2 λpτ1

τ1 τ1

τ1 τ2 a λpτ1 λpτ1 a τ2 τ1

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 22 / 25

It’s more complicated for winding arcs with multiple self-intersections.

×

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 23 / 25

It’s more complicated for winding arcs with multiple self-intersections.

×

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 23 / 25

It’s more complicated for winding arcs with multiple self-intersections.

×

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 23 / 25

It’s more complicated for winding arcs with multiple self-intersections.

×

For these arcs, we’re still codifying general rules for building snake graphs, but have conjectural snake graphs for some examples.

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 23 / 25

Example for p = 6: τ3 τ2 τ1

×

τ2 τ3 τ3 τ2 τ3 τ1 τ2

√ 3τ3

τ2

√ 3τ3

τ1 τ3 2τ3 2τ3

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 24 / 25

Questions?

Elizabeth Kelley Snake graphs from orbifolds April 14, 2019 25 / 25