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Kauffman brackets on surfaces
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Kauffman brackets on surfaces
Kauffman brackets on surfaces Francis Bonahon University of - - PowerPoint PPT Presentation
Kauffman brackets on surfaces Francis Bonahon University of - - PowerPoint PPT Presentation
Kauffman brackets on surfaces Kauffman brackets on surfaces Francis Bonahon University of Southern California Geometric Topology in New York, August 2013 1/28 Kauffman brackets on surfaces Joint work with Helen Wong 2/28 Kauffman brackets on
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Kauffman brackets on surfaces
Joint work with Helen Wong here with Grace Tsapsie Hibbard, born March 22, 2013
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Kauffman brackets on surfaces SL2(C)–characters
S = closed oriented surface of genus g 0 group homomorphism ρ: π1(S) → SL2(C)
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Kauffman brackets on surfaces SL2(C)–characters
S = closed oriented surface of genus g 0 group homomorphism ρ: π1(S) → SL2(C)
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Kauffman brackets on surfaces SL2(C)–characters
S = closed oriented surface of genus g 0 A group homomorphism ρ: π1(S) → SL2(C) defines its character Kρ : {closed curves in S} − → C K − → Tr ρ(K)
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Kauffman brackets on surfaces SL2(C)–characters
S = closed oriented surface of genus g 0 A group homomorphism ρ: π1(S) → SL2(C) defines its character Kρ : {closed multicurves in S} − → C K =
n
- i=1
Ki − → (−1)n
n
- i=1
Tr ρ(Ki)
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Kauffman brackets on surfaces SL2(C)–characters
Theorem (Helling 1967)
A function K: {closed multicurves in S} − → C is the character of a group homomorphism ρ: π1(S) → SL2(C) if and only if:
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Kauffman brackets on surfaces SL2(C)–characters
Theorem (Helling 1967)
A function K: {closed multicurves in S} − → C is the character of a group homomorphism ρ: π1(S) → SL2(C) if and only if:
◮ (Homotopy Invariance) K(K) depends only on the homotopy
class of K
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Kauffman brackets on surfaces SL2(C)–characters
Theorem (Helling 1967)
A function K: {closed multicurves in S} − → C is the character of a group homomorphism ρ: π1(S) → SL2(C) if and only if:
◮ (Homotopy Invariance) K(K) depends only on the homotopy
class of K
◮ (Superposition Rule) K(K1 ∪ K2) = K(K1)K(K2)
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Kauffman brackets on surfaces SL2(C)–characters
Theorem (Helling 1967)
A function K: {closed multicurves in S} − → C is the character of a group homomorphism ρ: π1(S) → SL2(C) if and only if:
◮ (Homotopy Invariance) K(K) depends only on the homotopy
class of K
◮ (Superposition Rule) K(K1 ∪ K2) = K(K1)K(K2) ◮ (Skein Relation) K(K1) = −K(K0) − K(K∞) if K1, K0, K∞
are the same everywhere, except in a small box where K1 = , K0 = and K∞ =
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Kauffman brackets on surfaces SL2(C)–characters
Theorem (Helling 1967)
A function K: {closed multicurves in S} − → C is the character of a group homomorphism ρ: π1(S) → SL2(C) if and only if:
◮ (Homotopy Invariance) K(K) depends only on the homotopy
class of K
◮ (Superposition Rule) K(K1 ∪ K2) = K(K1)K(K2) ◮ (Skein Relation) K(K1) = −K(K0) − K(K∞) if K1, K0, K∞
are the same everywhere, except in a small box where K1 = , K0 = and K∞ =
The Skein Relation just rephrases the classical trace relation of SL2(C): Tr M Tr N = Tr MN + Tr MN−1, ∀M, N ∈ SL2(C)
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Kauffman brackets on surfaces SL2(C)–characters
Definition
An SL2(C)–character is a function K: {closed multicurves in S} − → C such that:
◮ (Homotopy Invariance) K(K) depends only on the homotopy
class of K
◮ (Superposition Rule) K(K1 ∪ K2) = K(K1)K(K2) for any
multicurves K1 and K2
◮ (Skein Relation) K(K1) = −K(K0) − K(K∞) if K1, K0, K∞
are the same everywhere, except in a small box where K1 = , K0 = and K∞ =
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Kauffman brackets on surfaces Kauffman brackets
Definition
For q = e2πi ∈ C − {0}, a Kauffman q–bracket is a function K: {framed links in S × [0, 1]} − → End(E) for a finite-dimensional vector space E, such that:
◮ (Isotopy Invariance) K(K) depends only on the isotopy class
- f K in S × [0, 1]
◮ (Superposition Rule) K(K1 ∪ K2) = K(K1) ◦ K(K2) whenever
K = K1 ∪ K2 with K1 ⊂ S × [0, 1
2] and K2 ⊂ S × [1 2, 1] ◮ (Skein Relation) K(K1) = q
1 2 K(K0) + q− 1 2 K(K∞) if K1, K0,
K∞ are the same everywhere, except in a small box where K1 = , K0 = and K∞ =
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Kauffman brackets on surfaces Kauffman brackets
Definition
For q = e2πi ∈ C − {0}, a Kauffman q–bracket is a function K: {framed links in S × [0, 1]} − → End(E) for a finite-dimensional vector space E, such that:
◮ (Isotopy Invariance) K(K) depends only on the isotopy class
- f K in S × [0, 1]
◮ (Superposition Rule) K(K1 ∪ K2) = K(K1) ◦ K(K2) whenever
K = K1 ∪ K2 with K1 ⊂ S × [0, 1
2] and K2 ⊂ S × [1 2, 1] ◮ (Skein Relation) K(K1) = q
1 2 K(K0) + q− 1 2 K(K∞) if K1, K0,
K∞ are the same everywhere, except in a small box where K1 = , K0 = and K∞ =
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Kauffman brackets on surfaces Kauffman brackets
Definition
For q = e2πi ∈ C − {0}, a Kauffman q–bracket is a function K: {framed links in S × [0, 1]} − → End(E) for a finite-dimensional vector space E, such that:
◮ (Isotopy Invariance) K(K) depends only on the isotopy class
- f K in S × [0, 1]
◮ (Superposition Rule) K(K1 ∪ K2) = K(K1) ◦ K(K2) whenever
K = K1 ∪ K2 with K1 ⊂ S × [0, 1
2] and K2 ⊂ S × [1 2, 1] ◮ (Skein Relation) K(K1) = q
1 2 K(K0) + q− 1 2 K(K∞) if K1, K0,
K∞ are the same everywhere, except in a small box where K1 = , K0 = and K∞ =
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Kauffman brackets on surfaces Kauffman brackets
Definition
For q = e2πi ∈ C − {0}, a Kauffman q–bracket is a function K: {framed links in S × [0, 1]} − → End(E) for a finite-dimensional vector space E, such that:
◮ (Isotopy Invariance) K(K) depends only on the isotopy class
- f K in S × [0, 1]
◮ (Superposition Rule) K(K1 ∪ K2) = K(K1) ◦ K(K2) whenever
K = K1 ∪ K2 with K1 ⊂ S × [0, 1
2] and K2 ⊂ S × [1 2, 1] ◮ (Skein Relation) K(K1) = q
1 2 K(K0) + q− 1 2 K(K∞) if K1, K0,
K∞ are the same everywhere, except in a small box where K1 = , K0 = and K∞ =
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Kauffman brackets on surfaces Kauffman brackets
Definition
For q = e2πi ∈ C − {0}, a Kauffman q–bracket is a function K: {framed links in S × [0, 1]} − → End(E) ∼ = Mn(C) for a finite-dimensional vector space E, such that:
◮ (Isotopy Invariance) K(K) depends only on the isotopy class
- f K in S × [0, 1]
◮ (Superposition Rule) K(K1 ∪ K2) = K(K1) ◦ K(K2) whenever
K = K1 ∪ K2 with K1 ⊂ S × [0, 1
2] and K2 ⊂ S × [1 2, 1] ◮ (Skein Relation) K(K1) = q
1 2 K(K0) + q− 1 2 K(K∞) if K1, K0,
K∞ are the same everywhere, except in a small box where K1 = , K0 = and K∞ =
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Kauffman brackets on surfaces Kauffman brackets
Definition
For q = e2πi ∈ C − {0}, a Kauffman q–bracket is a function K: {framed links in S × [0, 1]} − → End(E) ∼ = Mn(C) for a finite-dimensional vector space E, such that:
◮ (Isotopy Invariance) K(K) depends only on the isotopy class
- f K in S × [0, 1]
◮ (Superposition Rule) K(K1 ∪ K2) = K(K1) ◦ K(K2) whenever
K = K1 ∪ K2 with K1 ⊂ S × [0, 1
2] and K2 ⊂ S × [1 2, 1] ◮ (Skein Relation) K(K1) = q
1 2 K(K0) + q− 1 2 K(K∞) if K1, K0,
K∞ are the same everywhere, except in a small box where K1 = , K0 = and K∞ =
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Kauffman brackets on surfaces Kauffman brackets
Definition
For q = e2πi ∈ C − {0}, a Kauffman q–bracket is a function K: {framed links in S × [0, 1]} − → End(E) ∼ = Mn(C) for a finite-dimensional vector space E, such that:
◮ (Isotopy Invariance) K(K) depends only on the isotopy class
- f K in S × [0, 1]
◮ (Superposition Rule) K(K1 ∪ K2) = K(K1) ◦ K(K2) whenever
K = K1 ∪ K2 with K1 ⊂ S × [0, 1
2] and K2 ⊂ S × [1 2, 1] ◮ (Skein Relation) K(K1) = q
1 2 K(K0) + q− 1 2K(K∞) if K1, K0,
K∞ are the same everywhere, except in a small box where K1 = , K0 = and K∞ =
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Kauffman brackets on surfaces Kauffman brackets Historic examples
- 1. When S = the sphere and End(E) = End(C) = C, the only
example is the classical Kauffman bracket (∼ = Jones polynomial) K: {framed links in R3} − → C
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Kauffman brackets on surfaces Kauffman brackets Historic examples
- 1. When S = the sphere and End(E) = End(C) = C, the only
example is the classical Kauffman bracket (∼ = Jones polynomial) K: {framed links in R3} − → C
- 2. Witten’s interpretation (1987) of the Jones polynomial in the
framework of a topological quantum field theory, mathematicalized by Reshetikhin-Turaev, provides a Kauffman q–bracket KWRT : {framed links in S × [0, 1]} − → End(E) for every q that is an N–root of unity with N odd.
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Kauffman brackets on surfaces Kauffman brackets Historic examples
- 1. When S = the sphere and End(E) = End(C) = C, the only
example is the classical Kauffman bracket (∼ = Jones polynomial) K: {framed links in R3} − → C
- 2. Witten’s interpretation (1987) of the Jones polynomial in the
framework of a topological quantum field theory, mathematicalized by Reshetikhin-Turaev, provides a Kauffman q–bracket KWRT : {framed links in S × [0, 1]} − → End(E) for every q that is an N–root of unity with N odd. The skein relation appears as a consequence of a property of the quantum trace in the quantum group Uq(sl2)
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Kauffman brackets on surfaces Kauffman brackets Historic examples
- 1. When S = the sphere and End(E) = End(C) = C, the only
example is the classical Kauffman bracket (∼ = Jones polynomial) K: {framed links in R3} − → C
- 2. Witten’s interpretation (1987) of the Jones polynomial in the
framework of a topological quantum field theory, mathematicalized by Reshetikhin-Turaev, provides a Kauffman q–bracket KWRT : {framed links in S × [0, 1]} − → End(E) for every q that is an N–root of unity with N odd. The skein relation appears as a consequence of a property of the quantum trace in the quantum group Uq(sl2) Goal of this talk: Construct other examples of Kauffman brackets
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Kauffman brackets on surfaces Kauffman brackets Conceptual motivation
When q = 1 and q
1 2 = −1, an irreducible Kauffman 1–bracket is
the same thing as an SL2(C)–character, namely as a point of the character variety RSL2(C)(S) = {homomorphisms ρ: π1(S) → SL2(C)}/ /SL2(C)
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Kauffman brackets on surfaces Kauffman brackets Conceptual motivation
When q = 1 and q
1 2 = −1, an irreducible Kauffman 1–bracket is
the same thing as an SL2(C)–character, namely as a point of the character variety RSL2(C)(S) = {homomorphisms ρ: π1(S) → SL2(C)}/ /SL2(C) Turaev (1987), Frohman, Bullock, Kania-Bartosz´ ynska, Przytycki, Sikora (around 2000): Interpretation of a Kauffman q–bracket as a “point” in a quantization of the character variety RSL2(C)(S), namely as a quantum SL2(C)–character.
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Kauffman brackets on surfaces Kauffman brackets Conceptual motivation
When q = 1 and q
1 2 = −1, an irreducible Kauffman 1–bracket is
the same thing as an SL2(C)–character, namely as a point of the character variety RSL2(C)(S) = {homomorphisms ρ: π1(S) → SL2(C)}/ /SL2(C) Turaev (1987), Frohman, Bullock, Kania-Bartosz´ ynska, Przytycki, Sikora (around 2000): Interpretation of a Kauffman q–bracket as a “point” in a quantization of the character variety RSL2(C)(S), namely as a quantum SL2(C)–character. From quantum to classical (Bonahon-Wong): When qN = 1 with N odd, every irreducible Kauffman q–bracket determines a character Kρ ∈ RSL2(C)(S), called the classical shadow of the Kauffman bracket.
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Kauffman brackets on surfaces Kauffman brackets Conceptual motivation
When q = 1 and q
1 2 = −1, an irreducible Kauffman 1–bracket is
the same thing as an SL2(C)–character, namely as a point of the character variety RSL2(C)(S) = {homomorphisms ρ: π1(S) → SL2(C)}/ /SL2(C) Turaev (1987), Frohman, Bullock, Kania-Bartosz´ ynska, Przytycki, Sikora (around 2000): Interpretation of a Kauffman q–bracket as a “point” in a quantization of the character variety RSL2(C)(S), namely as a quantum SL2(C)–character. From quantum to classical (Bonahon-Wong): When qN = 1 with N odd, every irreducible Kauffman q–bracket determines a character Kρ ∈ RSL2(C)(S), called the classical shadow of the Kauffman bracket. Today, from classical to quantum: Realize every character Kρ ∈ RSL2(C)(S) as the classical shadow of a Kauffman q–bracket.
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Kauffman brackets on surfaces Construction of SL2(C)–characters
How to construct a group homomorphism ρ: π1(S) → SL2(C)?
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Kauffman brackets on surfaces Construction of SL2(C)–characters
How to construct a group homomorphism ρ: π1(S) → SL2(C)? Pick a triangulation Γ of S, with vertex set VΓ
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Kauffman brackets on surfaces Construction of SL2(C)–characters
How to construct a group homomorphism ρ: π1(S) → SL2(C)? Pick a triangulation Γ of S, with vertex set VΓ Assign a weight xi ∈ C − {0} to each edge ei of Γ
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Kauffman brackets on surfaces Construction of SL2(C)–characters
How to construct a group homomorphism ρ: π1(S) → SL2(C)? Pick a triangulation Γ of S, with vertex set VΓ Assign a weight xi ∈ C − {0} to each edge ei of Γ This defines a pleated surface with shear-bend coordinates xi, and with monodromy ρ: π1(S − VΓ) → PSL2(C) = SL2(C)/ ± Id
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Kauffman brackets on surfaces Construction of SL2(C)–characters
How to construct a group homomorphism ρ: π1(S) → SL2(C)? Pick a triangulation Γ of S, with vertex set VΓ Assign a weight xi ∈ C − {0} to each edge ei of Γ This defines a pleated surface with shear-bend coordinates xi, and with monodromy ρ: π1(S − VΓ) → PSL2(C) = SL2(C)/ ± Id which, after choices of square roots x
1 2
i and of a spin structure,
defines a homomorphism ρ: π1(S − VΓ) → SL2(C)
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Kauffman brackets on surfaces Construction of SL2(C)–characters
How to construct a group homomorphism ρ: π1(S) → SL2(C)? Pick a triangulation Γ of S, with vertex set VΓ Assign a weight xi ∈ C − {0} to each edge ei of Γ This defines a pleated surface with shear-bend coordinates xi, and with monodromy ρ: π1(S − VΓ) → PSL2(C) = SL2(C)/ ± Id which, after choices of square roots x
1 2
i and of a spin structure,
defines a homomorphism ρ: π1(S − VΓ) → SL2(C) Main Point: The construction is classical and, for a curve K ⊂ S − VΓ, gives a very explicit formula for Tr ρ(K)
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Kauffman brackets on surfaces Construction of SL2(C)–characters
More precisely, if K crosses the edges ei1, ei2, . . . , ein,
Tr ρ(K) = ± Tr
- M1
- x
1 2
i1
x
− 1
2
i1
- M2
- x
1 2
i2
x
− 1
2
i2
- . . . Mn
- x
1 2
in
x
− 1
2
in
- where
Mk =
- 1
1 1
- if
- 1
1 1
- if
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Kauffman brackets on surfaces Construction of SL2(C)–characters
More precisely, if K crosses the edges ei1, ei2, . . . , ein,
Tr ρ(K) = ± Tr
- M1
- x
1 2
i1
x
− 1
2
i1
- M2
- x
1 2
i2
x
− 1
2
i2
- . . . Mn
- x
1 2
in
x
− 1
2
in
- = ±
- ±±···±
(0 or 1) x
± 1
2
i1
x
± 1
2
i2
. . . x
± 1
2
in
where
Mk =
- 1
1 1
- if
- 1
1 1
- if
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Kauffman brackets on surfaces Construction of SL2(C)–characters
Problem: This defines an SL2(C)–character on the punctured surface S − VΓ, not necessarily on the closed surface S
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Kauffman brackets on surfaces Construction of SL2(C)–characters
Problem: This defines an SL2(C)–character on the punctured surface S − VΓ, not necessarily on the closed surface S ei1 ei2 ei3 ein ein−1 K K ′
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Kauffman brackets on surfaces Construction of SL2(C)–characters
Problem: This defines an SL2(C)–character on the punctured surface S − VΓ, not necessarily on the closed surface S ei1 ei2 ei3 ein ein−1 K K ′ Tr ρ(K) = Tr ρ(K ′)?
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Kauffman brackets on surfaces Construction of SL2(C)–characters
Problem: This defines an SL2(C)–character on the punctured surface S − VΓ, not necessarily on the closed surface S ei1 ei2 ei3 ein ein−1 K K ′ Tr ρ(K) = Tr ρ(K ′)?
Fact
The edge weights xi define an SL2(C)–character on the closed surface S if and only if, for every vertex, x
1 2
i1 x
1 2
i1 . . . x
1 2
i1 = −1
1 + xi1 + xi1xi2 + xi1xi2xi3 + · · · + xi1xi2 . . . xin−1 = 0
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Kauffman brackets on surfaces Construction of SL2(C)–characters
Summary Recipe to construct SL2(C)–characters:
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Kauffman brackets on surfaces Construction of SL2(C)–characters
Summary Recipe to construct SL2(C)–characters:
- 1. Choose a weight x
1 2
i ∈ C − {0} for each edge ei of the
triangulation Γ
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Kauffman brackets on surfaces Construction of SL2(C)–characters
Summary Recipe to construct SL2(C)–characters:
- 1. Choose a weight x
1 2
i ∈ C − {0} for each edge ei of the
triangulation Γ
- 2. This defines an SL2(C)–character for the punctured surface
S − VΓ by an explicit formula Kρ(K) = ±
- ±±···±
(0 or 1) x
± 1
2
i1 x ± 1
2
i2
. . . x
± 1
2
in
SLIDE 44
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Kauffman brackets on surfaces Construction of SL2(C)–characters
Summary Recipe to construct SL2(C)–characters:
- 1. Choose a weight x
1 2
i ∈ C − {0} for each edge ei of the
triangulation Γ
- 2. This defines an SL2(C)–character for the punctured surface
S − VΓ by an explicit formula Kρ(K) = ±
- ±±···±
(0 or 1) x
± 1
2
i1 x ± 1
2
i2
. . . x
± 1
2
in
- 3. This character induces a character for the closed surface S if
and only if
- x
1 2
i1 x
1 2
i2 . . . x
1 2
in = −1
1 + xi1 + xi1xi2 + xi1xi2xi3 + · · · + xi1xi2 . . . xin−1 = 0 for each vertex
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Kauffman brackets on surfaces Construction of Kauffman brackets
Fix q ∈ C with qN = 1, N odd
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Kauffman brackets on surfaces Construction of Kauffman brackets
Fix q ∈ C with qN = 1, N odd Want to construct a Kauffman q–bracket K: {framed links in S × [0, 1]} − → End(E)
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Kauffman brackets on surfaces Construction of Kauffman brackets
Fix q ∈ C with qN = 1, N odd Want to construct a Kauffman q–bracket K: {framed links in S × [0, 1]} − → End(E) namely such that:
◮ (Isotopy Invariance) K(K) depends only on the isotopy class
- f K in S × [0, 1]
◮ (Superposition Rule) K(K1 ∪ K2) = K(K1) ◦ K(K2) whenever
K = K1 ∪ K2 with K1 ⊂ S × [0, 1
2] and K2 ⊂ S × [1 2, 1] ◮ (Skein Relation) K(K1) = q
1 2 K(K0) + q− 1 2 K(K∞) if K1, K0,
K∞ are the same everywhere, except in a small box where K1 = , K0 = and K∞ =
SLIDE 48
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Kauffman brackets on surfaces Construction of Kauffman brackets
Summary: Recipe to construct SL2(C)–characters
- 1. Choose a weight x
1 2
i ∈ C − {0} for each edge ei of the
triangulation Γ
- 2. This defines an SL2(C)–character for the punctured surface
S − VΓ by an explicit formula Kρ(K) = ±
- ±±···±
(0 or 1) x
± 1
2
i1 x ± 1
2
i2
. . . x
± 1
2
in
- 3. This character induces a character for the closed surface S if
and only if
- x
1 2
i1 x
1 2
i2 . . . x
1 2
in = −1
1 + xi1 + xi1xi2 + xi1xi2xi3 + · · · + xi1xi2 . . . xin−1 = 0 for each vertex
SLIDE 49
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Kauffman brackets on surfaces Construction of Kauffman brackets
Step 1. Choose an invertible operator (= matrix) X
1 2
i ∈ End(E) for
each edge ei of the triangulation Γ,
SLIDE 50
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Kauffman brackets on surfaces Construction of Kauffman brackets
Step 1. Choose an invertible operator (= matrix) X
1 2
i ∈ End(E) for
each edge ei of the triangulation Γ, for an appropriate finite-dimensional vector space E
SLIDE 51
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Kauffman brackets on surfaces Construction of Kauffman brackets
Step 1. Choose an invertible operator (= matrix) X
1 2
i ∈ End(E) for
each edge ei of the triangulation Γ, for an appropriate finite-dimensional vector space E and in such a way that X
1 2
i X
1 2
j = qX
1 2
j X
1 2
i
whenever ei ej
SLIDE 52
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Kauffman brackets on surfaces Construction of Kauffman brackets
Step 1. Choose an invertible operator (= matrix) X
1 2
i ∈ End(E) for
each edge ei of the triangulation Γ, for an appropriate finite-dimensional vector space E and in such a way that X
1 2
i X
1 2
j = qX
1 2
j X
1 2
i
whenever ei ej This is the same thing as a representation of the Chekhov-Fock algebra
- f the triangulation Γ (= quantum Teichm¨
uller space of the punctured surface S − VΓ)
SLIDE 53
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Kauffman brackets on surfaces Construction of Kauffman brackets
Step 1. Choose an invertible operator (= matrix) X
1 2
i ∈ End(E) for
each edge ei of the triangulation Γ, for an appropriate finite-dimensional vector space E and in such a way that X
1 2
i X
1 2
j = qX
1 2
j X
1 2
i
whenever ei ej This is the same thing as a representation of the Chekhov-Fock algebra
- f the triangulation Γ (= quantum Teichm¨
uller space of the punctured surface S − VΓ)
Proposition (FB + Xiaobo Liu, 2007, relatively easy)
If qN = 1 with N odd, smallest dimensional choices of such operators X
1 2
i ∈ End(E) are classified by
◮ edge weights xi ∈ C∗ such that X
N 2
i
= x
1 2
i IdE
◮ choices of N–roots for numbers x
1 2
i1 x
1 2
i2 . . . x
1 2
in ∈ C∗ associated to the
vertices
SLIDE 54
16/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Summary: Recipe to construct SL2(C)–characters
- 1. Choose a weight x
1 2
i ∈ C − {0} for each edge ei of the
triangulation Γ
- 2. This defines an SL2(C)–character for the punctured surface
S − VΓ by an explicit formula Kρ(K) = ±
- ±±···±
(0 or 1) x
± 1
2
i1 x ± 1
2
i2
. . . x
± 1
2
in
- 3. This character induces a character for the closed surface S if
and only if
- x
1 2
i1 x
1 2
i2 . . . x
1 2
in = −1
1 + xi1 + xi1xi2 + xi1xi2xi3 + · · · + xi1xi2 . . . xin−1 = 0 for each vertex
SLIDE 55
16/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Summary: Recipe to construct SL2(C)–characters
- 1. Choose a weight x
1 2
i ∈ C − {0} for each edge ei of the
triangulation Γ
- 2. This defines an SL2(C)–character for the punctured surface
S − VΓ by an explicit formula Kρ(K) = ±
- ±±···±
(0 or 1) x
± 1
2
i1 x ± 1
2
i2
. . . x
± 1
2
in
- 3. This character induces a character for the closed surface S if
and only if
- x
1 2
i1 x
1 2
i2 . . . x
1 2
in = −1
1 + xi1 + xi1xi2 + xi1xi2xi3 + · · · + xi1xi2 . . . xin−1 = 0 for each vertex
SLIDE 56
17/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Step 2.
Theorem (FB + Helen Wong, 2011)
Given operators X
1 2
i ∈ End(E) associated to the edges of the
triangulation Γ as in Step 1, there is an explicit formula K(K) =
- ±±···±
(0 or ± q ) X
± 1
2
i1
X
± 1
2
i2
. . . X
± 1
2
in
that defines a Kauffman q–bracket K: {framed links in (S − VΓ) × [0, 1]} − → End(E) for the punctured surface S − VΓ
SLIDE 57
17/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Step 2.
Theorem (FB + Helen Wong, 2011)
Given operators X
1 2
i ∈ End(E) associated to the edges of the
triangulation Γ as in Step 1, there is an explicit formula K(K) =
- ±±···±
(0 or ± q ) X
± 1
2
i1
X
± 1
2
i2
. . . X
± 1
2
in
that defines a Kauffman q–bracket K: {framed links in (S − VΓ) × [0, 1]} − → End(E) for the punctured surface S − VΓ Remark Much harder. Need to worry about the order in which to multiply the operators X
1 2
i ∈ End(E), which requires the
introduction of correction factors q related to the classical Kauffman bracket in R3.
SLIDE 58
17/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Step 2.
Theorem (FB + Helen Wong, 2011)
Given operators X
1 2
i ∈ End(E) associated to the edges of the
triangulation Γ as in Step 1, there is an explicit formula K(K) =
- ±±···±
(0 or ± q ) X
± 1
2
i1
X
± 1
2
i2
. . . X
± 1
2
in
that defines a Kauffman q–bracket K: {framed links in (S − VΓ) × [0, 1]} − → End(E) for the punctured surface S − VΓ Remark Much harder. Need to worry about the order in which to multiply the operators X
1 2
i ∈ End(E), which requires the
introduction of correction factors q related to the classical Kauffman bracket in R3. FB + Qingtao Chen, 2013 More conceptual approach based on the representation theory of the quantum group Uq(sl2)
SLIDE 59
18/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Summary: Recipe to construct SL2(C)–characters
- 1. Choose a weight x
1 2
i ∈ C − {0} for each edge ei of the
triangulation Γ
- 2. This defines an SL2(C)–character for the punctured surface
S − VΓ by an explicit formula Kρ(K) = ±
- ±±···±
(0 or 1) x
± 1
2
i1 x ± 1
2
i2
. . . x
± 1
2
in
- 3. This character induces a character for the closed surface S if
and only if
- x
1 2
i1 x
1 2
i2 . . . x
1 2
in = −1
1 + xi1 + xi1xi2 + xi1xi2xi3 + · · · + xi1xi2 . . . xin−1 = 0 for each vertex
SLIDE 60
18/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Summary: Recipe to construct SL2(C)–characters
- 1. Choose a weight x
1 2
i ∈ C − {0} for each edge ei of the
triangulation Γ
- 2. This defines an SL2(C)–character for the punctured surface
S − VΓ by an explicit formula Kρ(K) = ±
- ±±···±
(0 or 1) x
± 1
2
i1 x ± 1
2
i2
. . . x
± 1
2
in
- 3. This character induces a character for the closed surface S if
and only if
- x
1 2
i1 x
1 2
i2 . . . x
1 2
in = −1
1 + xi1 + xi1xi2 + xi1xi2xi3 + · · · + xi1xi2 . . . xin−1 = 0 for each vertex
SLIDE 61
19/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Problem: This defines a Kauffman bracket K: {framed links in (S − VΓ) × [0, 1]} − → End(E)
- n the punctured surface S − VΓ, not necessarily on the closed
surface S
SLIDE 62
19/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Problem: This defines a Kauffman bracket K: {framed links in (S − VΓ) × [0, 1]} − → End(E)
- n the punctured surface S − VΓ, not necessarily on the closed
surface S ei1 ei2 ei3 ein ein−1 K K ′
SLIDE 63
19/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Problem: This defines a Kauffman bracket K: {framed links in (S − VΓ) × [0, 1]} − → End(E)
- n the punctured surface S − VΓ, not necessarily on the closed
surface S ei1 ei2 ei3 ein ein−1 K K ′ K(K) = K(K ′)?
SLIDE 64
20/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Summary: Recipe to construct SL2(C)–characters
- 1. Choose a weight x
1 2
i ∈ C − {0} for each edge ei of the
triangulation Γ
- 2. This defines an SL2(C)–character for the punctured surface
S − VΓ by an explicit formula Kρ(K) = ±
- ±±···±
(0 or 1) x
± 1
2
i1 x ± 1
2
i2
. . . x
± 1
2
in
- 3. This character induces a character for the closed surface S if
and only if
- x
1 2
i1 x
1 2
i2 . . . x
1 2
in = −1
1 + xi1 + xi1xi2 + xi1xi2xi3 + · · · + xi1xi2 . . . xin−1 = 0 for each vertex
SLIDE 65
21/28
Kauffman brackets on surfaces Construction of Kauffman brackets
In Step 1, we associated to the edges of the triangulation Γ
- perators X
1 2
i ∈ End(E) such that X
N 2
i
= x
1 2
i IdE
SLIDE 66
21/28
Kauffman brackets on surfaces Construction of Kauffman brackets
In Step 1, we associated to the edges of the triangulation Γ
- perators X
1 2
i ∈ End(E) such that X
N 2
i
= x
1 2
i IdE
Step 3a. If x
1 2
i1 x
1 2
i2 . . . x
1 2
in = −1 at a vertex, the corresponding
- perators X
1 2
i ∈ End(E) can be chosen so that
X
1 2
i1 X
1 2
i2 . . . X
1 2
in = −q
n+2 4 IdE
SLIDE 67
22/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Summary: Recipe to construct SL2(C)–characters
- 1. Choose a weight x
1 2
i ∈ C − {0} for each edge ei of the
triangulation Γ
- 2. This defines an SL2(C)–character for the punctured surface
S − VΓ by an explicit formula Kρ(K) = ±
- ±±···±
(0 or 1) x
± 1
2
i1 x ± 1
2
i2
. . . x
± 1
2
in
- 3. This character induces a character for the closed surface S if
and only if
- x
1 2
i1 x
1 2
i2 . . . x
1 2
in = −1
1 + xi1 + xi1xi2 + xi1xi2xi3 + · · · + xi1xi2 . . . xin−1 = 0 for each vertex
SLIDE 68
22/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Summary: Recipe to construct SL2(C)–characters
- 1. Choose a weight x
1 2
i ∈ C − {0} for each edge ei of the
triangulation Γ
- 2. This defines an SL2(C)–character for the punctured surface
S − VΓ by an explicit formula Kρ(K) = ±
- ±±···±
(0 or 1) x
± 1
2
i1 x ± 1
2
i2
. . . x
± 1
2
in
- 3. This character induces a character for the closed surface S if
and only if
- x
1 2
i1 x
1 2
i2 . . . x
1 2
in = −1
1 + xi1 + xi1xi2 + xi1xi2xi3 + · · · + xi1xi2 . . . xin−1 = 0 for each vertex
SLIDE 69
23/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Step 3b. For a vertex v =
ei1 ei2 ei3 ein ein−1
- f the triangulation Γ for
the operators X
1 2
ij ∈ End(E) associated to the edges, consider
1+qXi1+q2Xi1Xi2+q3Xi1Xi2Xi3+ · · · +qn−1Xi1Xi2 . . . Xin−1
SLIDE 70
23/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Step 3b. For a vertex v =
ei1 ei2 ei3 ein ein−1
- f the triangulation Γ for
the operators X
1 2
ij ∈ End(E) associated to the edges, set
Fv = ker
- 1+qXi1+q2Xi1Xi2+q3Xi1Xi2Xi3+ · · · +qn−1Xi1Xi2 . . . Xin−1
SLIDE 71
23/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Step 3b. For a vertex v =
ei1 ei2 ei3 ein ein−1
- f the triangulation Γ for
the operators X
1 2
ij ∈ End(E) associated to the edges, set
Fv = ker
- 1+qXi1+q2Xi1Xi2+q3Xi1Xi2Xi3+ · · · +qn−1Xi1Xi2 . . . Xin−1
- and
F =
- vertices v
Fv ⊂ E
SLIDE 72
24/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Theorem
- 1. The linear subspace F ⊂ E is invariant under the image of the
Kauffman bracket K: {framed links in (S − VΓ) × [0, 1]} − → End(E) constructed in Step 2
SLIDE 73
24/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Theorem
- 1. The linear subspace F ⊂ E is invariant under the image of the
Kauffman bracket K: {framed links in (S − VΓ) × [0, 1]} − → End(E) constructed in Step 2 (but not invariant under the Xi!!)
SLIDE 74
24/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Theorem
- 1. The linear subspace F ⊂ E is invariant under the image of the
Kauffman bracket K: {framed links in (S − VΓ) × [0, 1]} − → End(E) constructed in Step 2 (but not invariant under the Xi!!)
- 2. If K, K ′ ⊂ (S − VΓ) × [0, 1] are isotopic in S × [0, 1], then
K(K)|F = K(K ′)|F
ei1 ei2 ei3 ein ein−1 K K ′
SLIDE 75
24/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Theorem
- 1. The linear subspace F ⊂ E is invariant under the image of the
Kauffman bracket K: {framed links in (S − VΓ) × [0, 1]} − → End(E) constructed in Step 2 (but not invariant under the Xi!!)
- 2. If K, K ′ ⊂ (S − VΓ) × [0, 1] are isotopic in S × [0, 1], then
K(K)|F = K(K ′)|F
ei1 ei2 ei3 ein ein−1 K K ′
Corollary
K induces a Kauffman q–bracket ¯ K: {framed links in S × [0, 1]} − → End(F) for the closed surface S
SLIDE 76
25/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Theorem
dim F N3(g−1) if g 2 N if g = 1 1 if g = 0
SLIDE 77
25/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Theorem
dim F N3(g−1) if g 2 N if g = 1 1 if g = 0
Theorem
Up to isomorphism, the Kauffman bracket ¯ K: {framed links in S × [0, 1]} − → End(F) depends only on the (classical) SL2(C)–character Kρ ∈ RSL2(C)(S) associated to the same edge weights xi ∈ C∗. In particular, it is independent of the triangulation Γ
SLIDE 78
25/28
Kauffman brackets on surfaces Construction of Kauffman brackets
Theorem
dim F N3(g−1) if g 2 N if g = 1 1 if g = 0 with equality for generic (all?) Kρ ∈ RSL2(C)(S)
Theorem
Up to isomorphism, the Kauffman bracket ¯ K: {framed links in S × [0, 1]} − → End(F) depends only on the (classical) SL2(C)–character Kρ ∈ RSL2(C)(S) associated to the same edge weights xi ∈ C∗. In particular, it is independent of the triangulation Γ
SLIDE 79
26/28
Kauffman brackets on surfaces Construction of Kauffman brackets From quantum to classical: the classical shadow
Theorem (Bonahon-Wong, 2012)
When qN = 1 with N odd, every irreducible Kauffman q–bracket K: {framed links in S × [0, 1]} − → End(E) determines a classical character Kρ ∈ RSL2(C)(S)
SLIDE 80
26/28
Kauffman brackets on surfaces Construction of Kauffman brackets From quantum to classical: the classical shadow
Theorem (Bonahon-Wong, 2012)
When qN = 1 with N odd, every irreducible Kauffman q–bracket K: {framed links in S × [0, 1]} − → End(E) determines a classical character Kρ ∈ RSL2(C)(S) Kρ : {closed multicurves in S} − → C
SLIDE 81
26/28
Kauffman brackets on surfaces Construction of Kauffman brackets From quantum to classical: the classical shadow
Theorem (Bonahon-Wong, 2012)
When qN = 1 with N odd, every irreducible Kauffman q–bracket K: {framed links in S × [0, 1]} − → End(E) determines a classical character Kρ ∈ RSL2(C)(S) Kρ : {closed multicurves in S} − → C by the property that K(K) ∈ End(E) for every knot K ⊂ S × [0, 1] whose projection to S has no crossing and whose framing is vertical.
SLIDE 82
26/28
Kauffman brackets on surfaces Construction of Kauffman brackets From quantum to classical: the classical shadow
Theorem (Bonahon-Wong, 2012)
When qN = 1 with N odd, every irreducible Kauffman q–bracket K: {framed links in S × [0, 1]} − → End(E) determines a classical character Kρ ∈ RSL2(C)(S) Kρ : {closed multicurves in S} − → C by the property that TN
- K(K)
- ∈ End(E)
for every knot K ⊂ S × [0, 1] whose projection to S has no crossing and whose framing is vertical. Here, TN(x) is the (normalized) N–th Chebyshev polynomial of the first type defined by 2 cos Nθ = TN(2 cos θ)
SLIDE 83
26/28
Kauffman brackets on surfaces Construction of Kauffman brackets From quantum to classical: the classical shadow
Theorem (Bonahon-Wong, 2012)
When qN = 1 with N odd, every irreducible Kauffman q–bracket K: {framed links in S × [0, 1]} − → End(E) determines a classical character Kρ ∈ RSL2(C)(S) Kρ : {closed multicurves in S} − → C by the property that TN
- K(K)
- = Kρ(K) IdE ∈ End(E)
for every knot K ⊂ S × [0, 1] whose projection to S has no crossing and whose framing is vertical. Here, TN(x) is the (normalized) N–th Chebyshev polynomial of the first type defined by 2 cos Nθ = TN(2 cos θ)
SLIDE 84
26/28
Kauffman brackets on surfaces Construction of Kauffman brackets From quantum to classical: the classical shadow
Theorem (Bonahon-Wong, 2012)
When qN = 1 with N odd, every irreducible Kauffman q–bracket K: {framed links in S × [0, 1]} − → End(E) determines a classical character Kρ ∈ RSL2(C)(S) Kρ : {closed multicurves in S} − → C by the property that TN
- K(K)
- = Kρ(K) IdE ∈ End(E)
for every knot K ⊂ S × [0, 1] whose projection to S has no crossing and whose framing is vertical. Here, TN(x) is the (normalized) N–th Chebyshev polynomial of the first type defined by 2 cos Nθ = TN(2 cos θ)
This is not the (normalized) N–th Chebyshev polynomial of the second type SN(x) is defined by sin Nθ = SN(2 cosθ) sin θ which usually occurs in the representation theory of SL2 and Uq(sl2)
SLIDE 85
27/28
Kauffman brackets on surfaces Construction of Kauffman brackets Miraculous cancelations
For the Kauffman q–bracket that we constructed, K(K) =
- ±±···±
(0 or ± q ) X
± 1
2
i1
X
± 1
2
i2
. . . X
± 1
2
in
where the matrices X
1 2
i ∈ End(E) are such that
X
1 2
i X
1 2
j = q
X
1 2
j X
1 2
i
and X
N 2
i
= xi IdE
SLIDE 86
27/28
Kauffman brackets on surfaces Construction of Kauffman brackets Miraculous cancelations
For the Kauffman q–bracket that we constructed, K(K) =
- ±±···±
(0 or ± q ) X
± 1
2
i1
X
± 1
2
i2
. . . X
± 1
2
in
where the matrices X
1 2
i ∈ End(E) are such that
X
1 2
i X
1 2
j = q
X
1 2
j X
1 2
i
and X
N 2
i
= xi IdE TN
- K(K)
- =
- −NkiN
(polynomial in q±1) X
± k1
2
i1
X
± k2
2
i2
. . . X
± kn
2
in
About Nn terms.
SLIDE 87
27/28
Kauffman brackets on surfaces Construction of Kauffman brackets Miraculous cancelations
For the Kauffman q–bracket that we constructed, K(K) =
- ±±···±
(0 or ± q ) X
± 1
2
i1
X
± 1
2
i2
. . . X
± 1
2
in
where the matrices X
1 2
i ∈ End(E) are such that
X
1 2
i X
1 2
j = q
X
1 2
j X
1 2
i
and X
N 2
i
= xi IdE Miraculous cancelations when qN = 1! TN
- K(K)
- = ±
- ±±···±
(0 or 1) X
± N
2
i1
X
± N
2
i2
. . . X
± N
2
in
At most 2n terms.
SLIDE 88
27/28
Kauffman brackets on surfaces Construction of Kauffman brackets Miraculous cancelations
For the Kauffman q–bracket that we constructed, K(K) =
- ±±···±
(0 or ± q ) X
± 1
2
i1
X
± 1
2
i2
. . . X
± 1
2
in
where the matrices X
1 2
i ∈ End(E) are such that
X
1 2
i X
1 2
j = q
X
1 2
j X
1 2
i
and X
N 2
i
= xi IdE Miraculous cancelations when qN = 1! TN
- K(K)
- = ±
- ±±···±
(0 or 1) X
± N
2
i1
X
± N
2
i2
. . . X
± N
2
in
= ±
- ±±···±
(0 or 1) x
± 1
2
i1 x ± 1
2
i2
. . . x
± 1
2
in
IdE At most 2n terms.
SLIDE 89
27/28
Kauffman brackets on surfaces Construction of Kauffman brackets Miraculous cancelations
For the Kauffman q–bracket that we constructed, K(K) =
- ±±···±
(0 or ± q ) X
± 1
2
i1
X
± 1
2
i2
. . . X
± 1
2
in
where the matrices X
1 2
i ∈ End(E) are such that
X
1 2
i X
1 2
j = q
X
1 2
j X
1 2
i
and X
N 2
i
= xi IdE Miraculous cancelations when qN = 1! TN
- K(K)
- = ±
- ±±···±
(0 or 1) X
± N
2
i1
X
± N
2
i2
. . . X
± N
2
in
= ±
- ±±···±
(0 or 1) x
± 1
2
i1 x ± 1
2
i2
. . . x
± 1
2
in
IdE = Kρ(K) IdE At most 2n terms.
SLIDE 90
28/28
Kauffman brackets on surfaces Construction of Kauffman brackets Miraculous cancelations
Corollary
The classical shadow of the Kauffman q–bracket K that we constructed is the character Kρ ∈ RSL2(C)(S) associated to the same edge weights xi as K
SLIDE 91
28/28
Kauffman brackets on surfaces Construction of Kauffman brackets Miraculous cancelations
Corollary
The classical shadow of the Kauffman q–bracket K that we constructed is the character Kρ ∈ RSL2(C)(S) associated to the same edge weights xi as K Current proof of miraculous cancelations
SLIDE 92
28/28
Kauffman brackets on surfaces Construction of Kauffman brackets Miraculous cancelations
Corollary
The classical shadow of the Kauffman q–bracket K that we constructed is the character Kρ ∈ RSL2(C)(S) associated to the same edge weights xi as K Current proof of miraculous cancelations Wishful thinking to guess
SLIDE 93
28/28
Kauffman brackets on surfaces Construction of Kauffman brackets Miraculous cancelations
Corollary
The classical shadow of the Kauffman q–bracket K that we constructed is the character Kρ ∈ RSL2(C)(S) associated to the same edge weights xi as K Current proof of miraculous cancelations Wishful thinking to guess Brute force to check
SLIDE 94
28/28
Kauffman brackets on surfaces Construction of Kauffman brackets Miraculous cancelations