Elisha Peterson- Trace Diagrams, Spin Networks, and Spaces of Graphs

8/3/2019 Elisha Peterson- Trace Diagrams, Spin Networks, and Spaces of Graphs

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IntroductionTrace Diagrams and Their Properties

Structure of the Coordinate RingCentral Functions

Trace Diagrams, Spin Networks, and Spaces of GraphsKAIST Geometric Topology Fair

Elisha Peterson

United States Military Academy

July 10, 2007

Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
http://find/http://goback/


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Fundamental Group Representations and the Coordinate Ring

Motivation I

Every representation Hom(, G)where G = SL(2,C) induces a map C given by taking the trace of(x). These elements areconjugation-invariant and reside in

both C[Gr

]G

and the coordinate ringof the character variety.

C [Gr]G

= C[]x

tr((x))

??

?

For this reason, the study of the structure ofthe character variety and its coordinate ringoften boils down to examining trace relations[described in detail in Lawtons first talk].This talk will describe a more geometric way todiscuss trace relations.



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Motivation I


both C[Gr

]G


C [Gr]G

= C[]x

tr((x))

??

?




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Motivation I


both C[Gr

]G


C [Gr]G

= C[]x

tr((x))

??

?




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Motivation II

We will develop an algebra which reflects both the structure arising fromthe fundamental group and that arising from its representation.We call the requisite algebra the Trace Diagram Algebra T2, whoseelements are a special class of graphs marked by elements of SL(2,C).

C [Gr]G

x

tr((x)) = G(t)

T2

t = F(x)

F G

The purpose of this talk is to describe F and G, hence providing analternate category for studying the coordinate ring of the character variety.This alternate category simplifies many calculations, and provides greaterintuition for both trace polynomials and their connection with thecharacter variety.



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Motivation II


C [Gr]G

x

tr((x)) = G(t)

T2

t = F(x)

F G




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Motivation II


C [Gr]G

x tr((x)) = G(t)

T2

t = F(x)

F G



I d i


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Defining the Functor GTrace Diagram RelationsComputing Trace Identities

Outline

1 IntroductionFundamental Group Representations and the Coordinate Ring

2 Trace Diagrams and Their Properties


3 Structure of the Coordinate Ring

The Functor FSurfaces and Character VarietiesBeyond Rank Two

4 Central Functions


I d i


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Definition of Trace Diagrams

DefinitionA 2-trace diagram t T2 is a graph drawn in a box whose edges aremarked by matrices in M22. All vertices have degree one and occur atthe bottom of the box (inputs), or at the top of the box (outputs). Thediagrams are in general position relative to a certain up direction.

A B

C

A

Note. A = A1.

We use general position to mean the following:

Each strand is an embedding;

Crossings and matrix markings are disjoint from

local extrema;Diagrams are equivalent if isotopic, provided theprevious condition remains true and local extremaare neither added nor removed.

Diagrams with compatible inputs/outputs may be composed by placingone atop another. This corresponds to composition of functions.


I t d ti


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A B

C

A

Note. A = A1.







I t d ti


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A B

C

A

Note. A = A1.







Introduction


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A B

C

A

Note. A = A1.







Introduction


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A B

C

A

Note. A = A1.







Introduction


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The Trace Diagram Functor

First we will construct, for V = C2 the standard representation ofSL(2,C),

T2G

Fun(V V

i V V

o),

where i is the number inputs and o the number of outputs.

Example

G A BC

A : V3 V5


Introduction


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Component Decomposition of Trace Diagrams

The functor G(t) for a given trace diagram t will be defined by piecingtogether the action of G on smaller components.

Proposition

Every strand of a trace diagram may be uniquely decomposed into thecomponents , , A , , and .

A B

C

A

Uniqueness comes from our restriction on general position.


IntroductionD fi h G


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Proposition


A B

C

A



IntroductionD fi i h F G


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Proposition


A B

C

A



IntroductionD fi i th F t G


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Trace Diagram Component Maps

Definition

Given v, w C2, A M22, and the standard basis {e1, e2} ofC2,

Identity G

: V V takes v v

Group Action G A : V V takes v AvPermutations G

: V V V V takes v w w v

Cap G : V V C takes

e1 e1 0

e1 e2 +1

e2 e1 1

e2 e2 0

or a b det[a b]

Cup G : C V V takes 1 e1 e2 e2 e1Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs

IntroductionD fi i th F t G


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Trace Diagrams and Their PropertiesStructure of the Coordinate Ring

Central Functions



Definition


Identity G

: V V takes v v



Cap G : V V C takes

e1 e1 0

e1 e2 +1

e2 e1 1

e2 e2 0

or a b det[a b]


IntroductionDefining the Functor G


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Central Functions



Definition


Identity G

: V V takes v v



Cap G : V V C takes

e1 e1 0

e1 e2 +1

e2 e1 1

e2 e2 0

or a b det[a b]




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Central Functions



Definition


Identity G

: V V takes v v



Cap G : V V C takes

e1 e1 0

e1 e2 +1

e2 e1 1

e2 e2 0

or a b det[a b]




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Central Functions



Definition


Identity G

: V V takes v v



Cap G : V V C takes

e1 e1 0

e1 e2 +1

e2 e1 1

e2 e2 0

or a b det[a b]




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Central Functions



Definition


Identity G

: V V takes v v



Cap G : V V C takes

e1 e1 0

e1 e2 +1

e2 e1 1

e2 e2 0

or a b det[a b]




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Central Functions


Group Invariance

Theorem

The image ofG lies in the set of multilinear functions Vi Vj whichare invariant with respect to simultaneous conjugation of all matrix

elements by any X SL(2,C). In other words, for all X SL(2,C)

X G(t(A1, . . . , Ar)) X = G(t(XA1X, . . . , XArX)).

Here, represents the action X (v w) = Xv Xw .

X A B

C

A X = XAX XBX

XCX

XAX

Remark. As a corollary, closed trace diagrams are invariant under

simultaneous conjugation in the matrix variables!


IntroductionT Di d Th i P i

Defining the Functor G


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Central Functions


Group Invariance

Theorem





X A B

C

A X = XAX XBX

XCX

XAX

Remark. As a corollary, closed trace diagrams are invariant undersimultaneous conjugation in the matrix variables!


IntroductionT Di d Th i P i



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Central Functions


Group Invariance

Theorem





X A B

C

A X = XAX XBX

XCX

XAX

Remark. As a corollary, closed trace diagrams are invariant undersimultaneous conjugation in the matrix variables!


IntroductionT Di d Th i P ti



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Central Functions

g u GTrace Diagram RelationsComputing Trace Identities

Group Invariance: Proof

Proof.It suffices to verify this property for the component maps. If

X =

x11 x12x21 x22

, then

X G( ) X = X G( ) takes

1 X (e1 e2 e2 e1)

= (x11e1 + x21e2) (x12e1 + x22e2)

(x12e1 + x22e2) (x11e1 + x21e2)

= (x11x22 x12x21)(e1 e2 e2 e1)= e1 e2 e2 e1 = G( ).

The proof for the cap is similar, while the calculations for the remainingcomponent maps are trivial.


IntroductionT e Di s d Thei P e ties



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Central Functions

g GTrace Diagram RelationsComputing Trace Identities



X =

x11 x12x21 x22

, then


1 X (e1 e2 e2 e1)

= (x11e1 + x21e2) (x12e1 + x22e2)

(x12e1 + x22e2) (x11e1 + x21e2)

= (x11x22 x12x21)(e1 e2 e2 e1)= e1 e2 e2 e1 = G( ).






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Central Functions

gTrace Diagram RelationsComputing Trace Identities



X =

x11 x12x21 x22

, then


1 X (e1 e2 e2 e1)

= (x11e1 + x21e2) (x12e1 + x22e2)

(x12e1 + x22e2) (x11e1 + x21e2)

= (x11x22 x12x21)(e1 e2 e2 e1)= e1 e2 e2 e1 = G( ).






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Central Functions

Trace Diagram RelationsComputing Trace Identities



X =

x11 x12x21 x22

, then


1 X (e1 e2 e2 e1)

= (x11e1 + x21e2) (x12e1 + x22e2)

(x12e1 + x22e2) (x11e1 + x21e2)

= (x11x22 x12x21)(e1 e2 e2 e1)= e1 e2 e2 e1 = G( ).






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Central Functions


1

IntroductionFundamental Group Representations and the Coordinate Ring

2 Trace Diagrams and Their PropertiesDefining the Functor G


3 Structure of the Coordinate RingThe Functor F

Surfaces and Character VarietiesBeyond Rank Two

4 Central Functions





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Central Functions


Trace Diagram Relations

Several diagrammatic relations arise from the component definitions, andit is easy to find two diagrams for which G(t1) = G(t2). For example:

Proposition

G = G Proof.

Let v = v1e1 + v2e2 C2. Then

G (v) = (v)=

(v e1 e2 v e2 e1)

= (v2)e2 (v1)e1 = v.



Defining the Functor GT Di R l i


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Central Functions




Proposition

G = G Proof.


G (v) = (v)=

(v e1 e2 v e2 e1)

= (v2)e2 (v1)e1 = v.



Defining the Functor GT Di R l i


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Central Functions




Proposition

G = G Proof.


G (v) = (v)=

(v e1 e2 v e2 e1)

= (v2)e2 (v1)e1 = v.



Defining the Functor GT Di R l ti


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Central Functions


The Fundamental Binor Identity

Proposition (Fundamental Binor Identity)

=

Proof.

1 represents the map a b b a

2 want to show that : a b a b b a

3 verify each element in the basis for C2 C2, e.g.

e2 e1

1

(e1 e2 e2 e1) = e2 e1 e1 e2

4 remaining basis elements work similarly

Remark. The binor identity provides a means of eliminating all crossings

in an SL(2,C

) trace diagram!



Defining the Functor GT Di R l ti


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g pStructure of the Coordinate Ring

Central Functions




=

Proof.




e2 e1

1

(e1 e2 e2 e1) = e2 e1 e1 e2



in an SL(2,C

) trace diagram!Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs


Defining the Functor GTrace Diagram Relations


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Central Functions




=

Proof.




e2 e1

1

(e1 e2 e2 e1) = e2 e1 e1 e2



in an SL(2,C





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Central Functions




=

Proof.




e2 e1

1

(e1 e2 e2 e1) = e2 e1 e1 e2



in an SL(2,C





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=

Proof.




e2 e1

1

(e1 e2 e2 e1) = e2 e1 e1 e2



in an SL(2,C





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=

Proof.




e2 e1

1

(e1 e2 e2 e1) = e2 e1 e1 e2



in an SL(2,C



S f C



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Matrices at Critical Points

Proposition (Critical Points)Matrices pass through critical points via

A A = det(A) and A A = det(A) .

Proof.

The definition det A = a11a22 a12a21 becomes the diagram

det(A) =e1 e2

A A

e1 e2

e1 e2

A A

e1 e2

=e1 e2

A A

e1 e2

=A A

e1 e2

.

Corollary

A

=A AA

= det(A)A

andA

=

1

det(A)A

.



S f h C di Ri



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Proof.


det(A) =e1 e2

A A

e1 e2

e1 e2

A A

e1 e2

=e1 e2

A A

e1 e2

=A A

e1 e2

.

Corollary

A

=A AA

= det(A)A

andA

=

1

det(A)A

.



S f h C di Ri



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Proof.


det(A) =e1 e2

A A

e1 e2

e1 e2

A A

e1 e2

=e1 e2

A A

e1 e2

=A A

e1 e2

.

Corollary

A

=A AA

= det(A)A

andA

=

1

det(A)A

.



St t f th C di t Ri



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gComputing Trace Identities




Proof.


det(A) =e1 e2

A A

e1 e2

e1 e2

A A

e1 e2

=e1 e2

A A

e1 e2

=A A

e1 e2

.

Corollary

A

=A AA

= det(A)A

andA

=

1

det(A)A

.



Str ct re of the Coordinate Ring



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Crossings at Local Extrema

Proposition

Show that the diagram may be well-defined.

Proof.Move the crossing away from the extremum:

= = = + .

Using = would have provided the same answer.

Remark: This makes the condition that crossings are disjoint from localextrema unnecessary.



Structure of the Coordinate Ring



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Crossings at Local Extrema

Proposition

Show that the diagram may be well-defined.

Proof.Move the crossing away from the extremum:

= = = + .

Using = would have provided the same answer.

Remark: This makes the condition that crossings are disjoint from localextrema unnecessary.






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Computing Trace Identities

Looping Relation

Proposition (Looping Relation)

Arcs may be wrapped around as follows:

= and consequentlyA

=A

= A

Proof.

Apply the algebraic definition or the binor identity in each case.






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Closed Diagrams: Trace Polynomials

Closed diagrams with matrices can be thought of as

Functions Gr C invariant under simultaneous conjugation;Trace polynomials.

Proposition (Trace)

A

=A

=tr

(A) = a11 + a22.

Corollary

= I = tr(I) = 2 and A A = 2 det(A).






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Proposition (Trace)

A

=A

=tr

(A) = a11 + a22.

Corollary

= I = tr(I) = 2 and A A = 2 det(A).




Defining the Functor GTrace Diagram RelationsC i T Id i i


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Proposition (Trace)

A

=A

=tr

(A) = a11 + a22.

Proof.

A = A I : 1 Ae1 e2 Ae2 e1

= (a11e1 + a21e2) e2 (a12e1 + a22e2) e1

a11 + a22.

CorollaryElisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs



Defining the Functor GTrace Diagram RelationsC i T Id i i


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Proposition (Trace)

A

=A

=tr

(A) = a11 + a22.

Corollary

= I = tr(I) = 2 and A A = 2 det(A).




Defining the Functor GTrace Diagram RelationsC ti T Id titi


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St uctu e o t e Coo d ate gCentral Functions


Summary of Diagram Rules

Diagram Rule (Summary)

B

A= AB = = +

= =A

= A

A = det(A) A A A = det(A) A A = det(A)

A = A = tr(A) = 2 A A = 2 det(A)




Defining the Functor GTrace Diagram RelationsC ti T Id titi


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gCentral Functions



2 Trace Diagrams and Their PropertiesDefining the Functor GTrace Diagram RelationsComputing Trace Identities

3 Structure of the Coordinate RingThe Functor FSurfaces and Character VarietiesBeyond Rank Two

4 Central Functions






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gCentral Functions


Example I

Example

Use the binor identity to reduce tr(A2B) = [A2B] = A2B .

Solution 1. Draw the diagram with crossings and apply the binor identity:

A A B =

This corresponds to

[A2B] = [A]2[B] 2det(A)[B] det(B)[A][AB] + det(A)[B]

= [A]2[B] det(A)[B] det(B)[A][AB].






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gCentral Functions


Example I

Example



A A B = A A B A A B

This corresponds to








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Central FunctionsComputing Trace Identities

Example I

Example



A A B = A A B A A B A A B A A B

This corresponds to








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Example I

Example



A A B = A A B A A B A A B A A B

This corresponds to





Structure of the Coordinate RingC l F



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Example II: The Commutator Relation

Proposition

tr(ABAB) = [ABAB] = [A]2 + [B]2 + [AB]2 [A][B][AB] 2.

Proof.

Represent the trace diagrammatically asB

A

A

B

= A A B B .

= +

= +

= + + + .Re-insert matrices and keep track of signs:

[ABAB] = [B]2 [A][B][AB] + [A]2 + [AB]2 2.

Use [AB] = [A][B] [AB] to obtain the desired relation.



Structure of the Coordinate RingC l F i



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Proposition


Proof.


A

A

B

= A A B B .

= +

= +


[ABAB] = [B]2 [A][B][AB] + [A]2 + [AB]2 2.




Structure of the Coordinate RingC t l F ti



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Proposition


Proof.


A

A

B

= A A B B .

= +

= +


[ABAB] = [B]2 [A][B][AB] + [A]2 + [AB]2 2.




Structure of the Coordinate RingC t l F ti



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Central Functionsp g


Proposition


Proof.


A

A

B

= A A B B .

= +

= +


[ABAB] = [B]2 [A][B][AB] + [A]2 + [AB]2 2.







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Proposition


Proof.


A

A

B

= A A B B .

= +

= +


[ABAB] = [B]2 [A][B][AB] + [A]2 + [AB]2 2.







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Example III: The 2 2 Characteristic Equation

The characteristic equation arises by replacing the eigenvalues in thecharacteristic polynomial 2 tr(A) + det(A) = 0 with the matrix.

Proposition

The binor identity implies the characteristic equation

A2 tr(A)A + det(A)I = 0.

Proof.

By the binor identity,

AA = A A AA

This last expression is A2 = A tr(A) det(A)I, the characteristicequation.


IntroductionTrace Diagrams and Their PropertiesStructure of the Coordinate Ring

Central Functions



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Central Functions




4 Central Functions



Central Functions



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Central Functions

The Category Morphism

The next task is to construct F in the figure below.

C [Gr]G

x tr((x)) = G(t)

T2

t = F(x)

F G

We will use this construction to examine the structure of the coordinate

ring C[].



Central Functions



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Central Functions

Surface Group Representations

We now define the function F : 1 T2 which assigns a trace diagram toeach element of the fundamental group.

Assign surface cuts to

elements of thefundamental group.

Mark loops at the cutsusing the representation1 G.

Ensure drawing iscompatible with an updirection.

The additional action of G takes this element to tr(CBBA).



Central Functions



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Central Functions










Central Functions



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Central Functions







A

B

B

C




Central Functions



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C u







A

B

B

C




Central Functions



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A

B

B

C




Central Functions



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The Character Variety

The ring of trace polynomials may be used to construct the following:

The G-character varietyX is the algebraic variety whose coordinatering is the trace ring generated by representations.

In other words, the space of trace diagrams on a surface can be thoughtof as precisely C[X].



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Central Functions



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Rank Two

Proposition

For surfaces with free group of rank two, the coordinate ringC[] is apolynomial ring in three indeterminates.

Proof.

Given the binor identity

= , all trace loops onthe three-holed sphere can be

reduced to three basic looptypes.

A BA B



Central Functions



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4 Central Functions



Central Functions



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Beyond Crossing Removal

What happens when we remove all crossings in higher rank cases??

Problem. There are an infinite number of diagams without crossings!

Definition

A 2-trace diagram in a surface is simple if it has no self-crossings andpasses through each cut set (or contains each matrix variable) at mostonce.



Central Functions



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Definition




Central Functions



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Definition




Central Functions



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Definition




Central Functions



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Definition




Central Functions



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Definition




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Central Functions



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Reduction of Diagrams I

PropositionTrace diagram relations can reduce any surface loop to simple diagrams.

Proof.

Use the binor identity to remove crossings. For duplicates:

= =

=

= + .



Central Functions



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Reduction of Diagrams II

Remark. Algebraically, the ability to reducec is simply the statement thatSL(2,C) trace relations can be used to reduce the polynomials fortr(A A ) and tr(A A ).



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Diagrammatic Generators

We have now proven:

Theorem

The set of simple trace diagrams with no more than three elements

generates the space of all closed trace diagrams on a surface.

But there are more diagrammatic relations...






89/107

Diagrammatic Generators

We have now proven:

Theorem

The set of simple trace diagrams with no more than three elements

generates the space of all closed trace diagrams on a surface.

But there are more diagrammatic relations...






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Three Element Trace Relations I

Proposition

[ABC] + [CBA] = [A][BC] + [B][AC] + [C][AB] [A][B][C]

Proof.

The anti-symmetrizer 3 sends any a b c C2 C2 C2 to zero.

Given that 3 = 16

+ +

, this implies

the relations+ = + + + ;

A B C + A B C = A B C + A B C + A B C + A B C . This

is precisely the identity above.





Th El T R l i I


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Proposition


Proof.


Given that 3 = 16

+ +

, this implies








Th El T R l i I


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Proposition


Proof.


Given that 3 = 16

+ +

, this implies






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Th El T R l i II


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Three Element Trace Relations II

Proposition

[ABC][CBA] = [A]2 + [B]2 + [C]2 + [AB]2 + [BC]2 + [AC]2

[A][B][AB] [B][C][BC] [A][C][AC] + [AB][BC][AC] 4.

Proof.

The crux of the argument comes in reducing the crossings of the followingdiagram, which when closed gives the product [ABC][CBA]:

A A B B C C

The result will be a diagam with sixteen terms including loops forelements such as [AB] or [AB]. Applying the relation[AB] = [A][B] [AB] reduces the result to the above form.





Th El t T R l ti II


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Three Element Trace Relations II

Proposition

[ABC][CBA] = [A]2 + [B]2 + [C]2 + [AB]2 + [BC]2 + [AC]2

[A][B][AB] [B][C][BC] [A][C][AC] + [AB][BC][AC] 4.

Proof.

The crux of the argument comes in reducing the crossings of the followingdiagram, which when closed gives the product [ABC][CBA]:

A A B B C C

The result will be a diagam with sixteen terms including loops forelements such as [AB] or [AB]. Applying the relation[AB] = [A][B] [AB] reduces the result to the above form.





S I i t Th R lt


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Some Invariant Theory Results

Theorem

A minimal generating set of C[] consists of{tr(Xi)}, {tr(XiXj)} fori < j, and {tr(XiXjXk)} for i < j < k.

Theorem

A maximal independent set of generators consists of {tr(Xi)} and{tr(XiXj)} for j = i + 1 or j = i + 2.

We have diagrammatically proven these theorems, except fordemonstrating certain two-element trace relations.





S I i t Th R lt


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Some Invariant Theory Results

Theorem

A minimal generating set of C[] consists of{tr(Xi)}, {tr(XiXj)} fori < j, and {tr(XiXjXk)} for i < j < k.

Theorem

A maximal independent set of generators consists of {tr(Xi)} and{tr(XiXj)} for j = i + 1 or j = i + 2.

We have diagrammatically proven these theorems, except fordemonstrating certain two-element trace relations.





Wh th Di ti A h?


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Why the Diagrammatic Approach?

Diagrams are good for:

exhibiting mathematical structure

duality corresponds to turning diagrams upside-down

relations are often simply expressed: =

connecting algebra with geometrywhen placed on surfaces, trace diagrams describe the moduli space ofrepresentations of a surface group

discovering similarities among mathematical structures

= is both a 2 2 trace identity and the defining relation of

the Poisson bracket on the coordinate ring of the character varietycomputational algorithms

relations used to generate recurrence equations;illustrative method for generating trace identities.





Wh th Di g ti A h?


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Introduction


Central Functions




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Introduction


Central Functions


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1 Introduction


2 Trace Diagrams and Their PropertiesDefining the Functor GTrace Diagram Relations


3 Structure of the Coordinate RingThe Functor FSurfaces and Character Varieties

Beyond Rank Two

4 Central Functions


Introduction


Central Functions

Basis for the Coordinate Ring I


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Diagrams can be used to construct a basis for the trace ring.

AA BA BA BA BA BA B

76

5

Expand symmetrizers and remove crossings toobtain a trace polynomial

7,6

5

(tr(A), tr(B), tr(AB))

Theorem

The polynomials a,bc

comprise a basis for the

coordinate ring of theSL(2,C)-charactervariety of the

three-holed sphere.

Proof uses theunitary trick andthe Peter-WeylTheorem.


Introduction


Central Functions



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AA BA BA BA BA BA B

76

5


7,65 (tr(A), tr(B), tr(AB))

Theorem




three-holed sphere.



Introduction


Central Functions



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AA BA BA BA BA BA B

76

5


7,65 (tr(A), tr(B), tr(AB))

Theorem




three-holed sphere.



Introduction


Central Functions

Basis for the Coordinate Ring II


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Basis for the Coordinate Ring II

Shorthand:AA BA BA BA BA BA B

7 6

5

7

A

6

B5

Remarks:

Edges are labeled by representations.

The basis exhibits considerable symmetry.

The basis depends only on fundamental group of the surface.

To generalize for other surfaces, add more loops:A B C


Introduction


Central Functions

Acknowledgments/References


107/107

Acknowledgments/References

Bill Goldman

Sean Lawton

Charles Frohman, Doug Bullock, Joanna Kania-Bartoszynska.

Carter/Flath/Saito, The Classical and Quantum 6j-Symbols

Documents

Elisha Peterson- Trace Diagrams, Spin Networks, and Spaces of Graphs