# 講義の目標

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## 結び目の数学

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2

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!

2

C

0

1

0

1

!

C

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0

1

0

1

0

1

0

i

i,0

1

i

i,1

C

C

0

1

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C

1

2

1

2

1

2

1

2

i

j

i

i

i

j

j

j

i,j

i

j

i

j

(9)

i

j

i

i

i

j

j

j

i,j

i

j

i

j

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C

### ( C , C ) = C

%−→F

F( F( F(

F(

)⊗F( )⊗F( )⊗F(

)⊗F( ) )⊗F( )⊗F( )

) )

C C

V VV V

V V VV

V VV V

C C

F( ) : C→V ⊗V, F( ) : V ⊗V →C, F( ) : V ⊗V →V⊗V, F( ) = idV: V →V

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## RI での不変性の証明

F% &

=F% &

(idV ⊗F( ))◦(F( )⊗idV)◦(idV ⊗F( )) = idV

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## RII での不変性の証明

F% &

=F% &

F( )◦F( ) = idV ⊗idV

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## RIII での不変性の証明

F% &

=F% &

(F( )⊗idV)◦(idV ⊗F( ))◦(F( )⊗idV)

= (idV ⊗F( ))◦(F( )⊗idV)◦(idV ⊗F( ))

⇐⇒ F( )

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1

1/2

1/2

3/2

1/2

1

1/2

2

1/2

3/2

1

1

1/2

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### 量子不変量としてのジョーンズ多項式

F( ) =ev : V⊗V →C, f ⊗x%→f(x) F( ) =ev: V ⊗V →C, x⊗f %→f(h(x)) F( ) =coev : C→V ⊗V, 1%→v0⊗v0+v1⊗v1

F( ) =coev: C→V⊗V, 1%→v0⊗h−1(v0) +v1⊗h−1(v1)

h=

!t1/2 0

0 t1/2

"

### と置いた．

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1"−−−−−−−−−→coev∗⊗coev (t1/2v0v0+t1/2v1v1)(v0v0+v1v1)

=t1/2v0v0v0v0+t1/2v0v0v1v1 +t−1/2v1v1v0v0+t−1/2v1v1v1v1

idV∗ ⊗R⊗idV

"−−−−−−−−−−−−→t1/2v0(t1/2v0v0)v0+t1/2v0(tv1v0)v1

+t1/2v1(tv0v1+ (t1/2t3/2)v1v0)v0+t1/2v1(t1/2v1v1)v1 idV∗ ⊗RidV

"−−−−−−−−−−−−→t1/2v0t1/2(t1/2v0v0)v0+t1/2v0t(tv0v1+ (t1/2t3/2)v1v0)v1 +t1/2v1(t(tv1v0) + (t1/2t3/2)(tv0v1+ (t1/2t3/2)v1v0))v0

+t−1/2v1t1/2(t1/2v1v1)v1

=t3/2v0v0v0v0+t5/2v0v0v1v1+t1/2(t1/2t3/2)v0v1v0v1 +t1/2(t2+ (t1/2t3/2)2)v1v1v0v0+t1/2t(t1/2t3/2)v1v0v1v0 +t1/2v1v1v1v1

ev⊗ev

"−−−−−−→t3/2t1/2+t5/2t1/2+t1/2(t2+ (t1/2t3/2)2)t1/2+t1/2t1/2

= 1 +t+t2+t3

1/(t1/2 +t−1/2)

"−−−−−−−−−−−−−−→t1/2+t5/2

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2

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C

### ( C , C ) = C

%−→F

F( F( F(

F(

)⊗F( )⊗F( )⊗F(

)⊗F( ) )⊗F( )⊗F( )

) )

C C

V VWW

V WVW

V VWW

C C

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sl2

Uh(sl2)

sl2

J(L)∈Uh(sl2)⊗n

Uh(sl2)

n

Vn (n∈N)

### 色付きジョーンズ多項式

JVi1,...,Vin(L)∈C.

⇒⇒

L

m

Vim

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!

2

!

!

1

1/2

1/2

H/2

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h

2

h

2

V

h

2

Q

V

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h

2

## ), R, θ)

h

2

ˆ2

h

2

h

2

1

V

W

V

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### 色付きジョーンズ多項式

F( ) =τ ◦(ρV ⊗ρW)(R),: V ⊗W →W ⊗V, F( ) = (ρV ⊗ρV)(R1)◦τ: W ⊗V →V ⊗W, F( ) = ev : V⊗V →C, f ⊗x%→f(x)

F( ) = ev: V ⊗V →C, x⊗f %→f(ρ(K−1)(x)) F( ) = coev : C→V ⊗V, 1%→v0⊗v0 +v1⊗v1

F( ) = coev: C→V⊗V,1%→v0⊗ρ(K)(v0)+v1⊗ρ(K)(v1)

### ただし

τ: V ⊗V →V⊗V, a⊗b %→b⊗a.

V1,...,Vn

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2

H4H!

n0

12n(n1)

n

q

n

n

2

!

2

ˆn

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2

### J ( )

=qH⊗H2 ! \$

m,n0

q12m(m1)+12n(n1)+m2(q1)m+n

[m]q![n]q! FmKmEnEmKmFn

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2

1

n

!

2

Vi

Vi

!

2

i

i

V1,...,Vn

V1

Vn

2

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!

!

!

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## RI での不変性の証明

F% &

=F% &

(idV ⊗ev)◦(r⊗idV)◦(idV ⊗coev) = idV

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F% &

=F% &

r◦r1 = idV ⊗idV

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## RIII での不変性の証明

F% &

=F% &

(r⊗idV)◦(idV ⊗r)◦(r⊗idV)

= (idV ⊗r)◦(r⊗idV)◦(idV ⊗r)

⇐⇒ r

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1

1/2

1/2

1

1

1/2

1/2

V

V

C

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## References

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