8/3/2019 Elisha Peterson- Trace Diagrams, Spin Networks, and Spaces of Graphs
1/107
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
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2/107
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
http://find/http://goback/8/3/2019 Elisha Peterson- Trace Diagrams, Spin Networks, and Spaces of Graphs
3/107
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
http://find/http://goback/8/3/2019 Elisha Peterson- Trace Diagrams, Spin Networks, and Spaces of Graphs
4/107
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
http://find/http://goback/8/3/2019 Elisha Peterson- Trace Diagrams, Spin Networks, and Spaces of Graphs
5/107
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
Fundamental Group Representations and the Coordinate Ring
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
http://find/http://goback/8/3/2019 Elisha Peterson- Trace Diagrams, Spin Networks, and Spaces of Graphs
6/107
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
Fundamental Group Representations and the Coordinate Ring
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
http://find/http://goback/8/3/2019 Elisha Peterson- Trace Diagrams, Spin Networks, and Spaces of Graphs
7/107
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
Fundamental Group Representations and the Coordinate Ring
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
I d i
http://find/http://goback/8/3/2019 Elisha Peterson- Trace Diagrams, Spin Networks, and Spaces of Graphs
8/107
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
Outline
1 IntroductionFundamental Group Representations and the Coordinate Ring
2 Trace Diagrams and Their Properties
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
3 Structure of the Coordinate Ring
The Functor FSurfaces and Character VarietiesBeyond Rank Two
4 Central Functions
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
I d i
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9/107
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
I t d ti
http://find/http://goback/8/3/2019 Elisha Peterson- Trace Diagrams, Spin Networks, and Spaces of Graphs
10/107
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
I t d ti
http://find/http://goback/8/3/2019 Elisha Peterson- Trace Diagrams, Spin Networks, and Spaces of Graphs
11/107
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
Introduction
http://find/http://goback/8/3/2019 Elisha Peterson- Trace Diagrams, Spin Networks, and Spaces of Graphs
12/107
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
Introduction
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13/107
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
Introduction
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14/107
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
Introduction
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IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionD fi h G
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16/107
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionD fi i h F G
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17/107
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionD fi i th F t G
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18/107
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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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19/107
Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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
IntroductionDefining the Functor G
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Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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
IntroductionDefining the Functor G
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Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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
IntroductionDefining the Functor G
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22/107
Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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
IntroductionDefining the Functor G
http://find/http://goback/8/3/2019 Elisha Peterson- Trace Diagrams, Spin Networks, and Spaces of Graphs
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Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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
IntroductionDefining the Functor G
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Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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!
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionT Di d Th i P i
Defining the Functor G
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Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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 undersimultaneous conjugation in the matrix variables!
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionT Di d Th i P i
Defining the Functor G
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Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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 undersimultaneous conjugation in the matrix variables!
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionT Di d Th i P ti
Defining the Functor G
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Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionT e Di s d Thei P e ties
Defining the Functor G
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Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
g 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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Defining the Functor G
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29/107
Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Defining the Functor G
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30/107
Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
Trace 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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Defining the Functor G
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31/107
Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
Trace Diagram RelationsComputing Trace Identities
1
IntroductionFundamental Group Representations and the Coordinate Ring
2 Trace Diagrams and Their PropertiesDefining the Functor G
Trace Diagram RelationsComputing Trace Identities
3 Structure of the Coordinate RingThe Functor F
Surfaces and Character VarietiesBeyond Rank Two
4 Central Functions
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Defining the Functor G
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32/107
Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
Trace Diagram RelationsComputing Trace Identities
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Defining the Functor GT Di R l i
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Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
Trace Diagram RelationsComputing Trace Identities
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Defining the Functor GT Di R l i
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34/107
Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
Trace Diagram RelationsComputing Trace Identities
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Defining the Functor GT Di R l ti
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Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
Trace Diagram RelationsComputing Trace Identities
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!
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Defining the Functor GT Di R l ti
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g pStructure of the Coordinate Ring
Central Functions
Trace Diagram RelationsComputing Trace Identities
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!Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Defining the Functor GTrace Diagram Relations
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g pStructure of the Coordinate Ring
Central Functions
Trace Diagram RelationsComputing Trace Identities
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!Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Defining the Functor GTrace Diagram Relations
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g pStructure of the Coordinate Ring
Central Functions
Trace Diagram RelationsComputing Trace Identities
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!Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Defining the Functor GTrace Diagram Relations
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Structure of the Coordinate RingCentral Functions
Trace Diagram RelationsComputing Trace Identities
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!Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Defining the Functor GTrace Diagram Relations
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Structure of the Coordinate RingCentral Functions
Trace Diagram RelationsComputing Trace Identities
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!Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
S f C
Defining the Functor GTrace Diagram Relations
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Structure of the Coordinate RingCentral Functions
Trace Diagram RelationsComputing Trace Identities
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
.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
S f h C di Ri
Defining the Functor GTrace Diagram Relations
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Structure of the Coordinate RingCentral Functions
Trace Diagram RelationsComputing Trace Identities
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
.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
S f h C di Ri
Defining the Functor GTrace Diagram Relations
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Structure of the Coordinate RingCentral Functions
Trace Diagram RelationsComputing Trace Identities
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
.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
St t f th C di t Ri
Defining the Functor GTrace Diagram Relations
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Structure of the Coordinate RingCentral Functions
gComputing Trace Identities
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
.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Str ct re of the Coordinate Ring
Defining the Functor GTrace Diagram Relations
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Structure of the Coordinate RingCentral Functions
gComputing Trace Identities
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate Ring
Defining the Functor GTrace Diagram Relations
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Structure of the Coordinate RingCentral Functions
gComputing Trace Identities
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate Ring
Defining the Functor GTrace Diagram Relations
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Structure of the Coordinate RingCentral Functions
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate Ring
Defining the Functor GTrace Diagram Relations
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Structure of the Coordinate RingCentral Functions
Computing Trace Identities
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).
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate Ring
Defining the Functor GTrace Diagram Relations
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Structure of the Coordinate RingCentral Functions
Computing Trace Identities
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).
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate Ring
Defining the Functor GTrace Diagram RelationsC i T Id i i
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Structure of the Coordinate RingCentral Functions
Computing Trace Identities
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.
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
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate Ring
Defining the Functor GTrace Diagram RelationsC i T Id i i
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Structure of the Coordinate RingCentral Functions
Computing Trace Identities
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).
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate Ring
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
Computing Trace Identities
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)
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate Ring
Defining the Functor GTrace Diagram RelationsC ti T Id titi
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gCentral Functions
Computing Trace Identities
1 IntroductionFundamental Group Representations and the Coordinate Ring
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
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate Ring
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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gCentral Functions
Computing Trace Identities
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].
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate Ring
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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gCentral Functions
Computing Trace Identities
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 = A A B 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].
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate Ring
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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Central FunctionsComputing Trace Identities
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 = A A B A A B A A B 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].
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate Ring
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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Central FunctionsComputing Trace Identities
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 = A A B A A B A A B 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].
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingC l F
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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Central FunctionsComputing Trace Identities
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingC l F i
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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Central FunctionsComputing Trace Identities
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingC t l F ti
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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Central FunctionsComputing Trace Identities
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingC t l F ti
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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Central Functionsp g
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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Central Functionsp g
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
Defining the Functor GTrace Diagram RelationsComputing Trace Identities
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Central Functionsp g
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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Central Functions
1 IntroductionFundamental Group Representations and the Coordinate Ring
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
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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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[].
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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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).
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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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).
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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68/107
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.
A
B
B
C
The additional action of G takes this element to tr(CBBA).
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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C u
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.
A
B
B
C
The additional action of G takes this element to tr(CBBA).
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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70/107
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.
A
B
B
C
The additional action of G takes this element to tr(CBBA).
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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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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IntroductionTrace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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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
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IntroductionTrace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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1 IntroductionFundamental Group Representations and the Coordinate Ring
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
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
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IntroductionTrace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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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:
= =
=
= + .
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IntroductionTrace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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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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IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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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...
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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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...
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
Th El T R l i I
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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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
Th El T R l i I
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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.
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IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
IntroductionTrace Diagrams and Their Properties
Structure of the Coordinate RingCentral Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
Wh th Di g 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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
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Introduction
Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
The Functor FSurfaces and Character VarietiesBeyond Rank Two
Why the Diagrammatic Approach?
8/3/2019 Elisha Peterson- Trace Diagrams, Spin Networks, and Spaces of Graphs
101/107
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
Introduction
Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
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102/107
1 Introduction
Fundamental Group Representations and the Coordinate Ring
2 Trace Diagrams and Their PropertiesDefining the Functor GTrace Diagram Relations
Computing Trace Identities
3 Structure of the Coordinate RingThe Functor FSurfaces and Character Varieties
Beyond Rank Two
4 Central Functions
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
Introduction
Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
Basis for the Coordinate Ring I
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Basis for the Coordinate Ring I
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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
Introduction
Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
Basis for the Coordinate Ring I
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Basis for the Coordinate Ring I
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
7,65 (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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
Introduction
Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
Basis for the Coordinate Ring I
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Basis for the Coordinate Ring I
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
7,65 (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.
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
Introduction
Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
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
Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
Introduction
Trace Diagrams and Their PropertiesStructure of the Coordinate Ring
Central Functions
Acknowledgments/References
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Acknowledgments/References
Bill Goldman
Sean Lawton
Charles Frohman, Doug Bullock, Joanna Kania-Bartoszynska.
Carter/Flath/Saito, The Classical and Quantum 6j-Symbols