On the Casson-Walker invariant and a quantum representation of the mapping class group through the LMO invariant for genus one open books Atsushi MOCHIZUKI

On the Casson-Walker invariant and a quantum representation of the mapping class group through

the LMO invariant for genus one open books

Atsushi MOCHIZUKI

RIMS, Kyoto University. D3 revised on 22th February, 2017

A table of contents

Introduction

• Backgrounds

• Motivations

• Def (Open book decomp) and Prop (Surgery presentation)

• Results Part A

• Def (Surgery) and Explanation (Surgery presentation)

• Idea of Proof of Thm A

• Definitions (Jacobi diag, Kontsevich inv, LMO inv)

• Proof of Thm A Part B

• Definitions (Kirby moves, Space of Jacobi diags, Product)

• Construction of Representation

• Proof of Thm B

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Backgrounds

Casson-Walker invλ(M)∈Q

• for aZ_::::HSM (Casson, 1980s) H_∗(M;Z)∼=H_∗(S³;Z)

• for aQ_::::HSM (Walker, 1992) H_∗(M;Q)∼=H_∗(S³;Q)

Construction in terms of mapping class groups with regard toHeegaard splittings

• Morita (1980s)

• Cheptea, Habiro, Massuyeau (2008) etc.

⇝How about the case of open book decompositions

Motivation A

g= 1 Heegaard splitting lens spacesL(p, q)

λ(L(p, q)) =−1 2s(q, p) (s(q, p) : Dedekind sum)

g= 1open book decomposition g= 1open books M_φ

(φ: homeo ofΣ_1,1)

λ(Mφ) =?

Aim A

Calculate the Casson-Walker invariant for 3-manifolds admitting a genus one open book decomposition.

4 / 34

Motivation B

quantum inv. of links

quantum rep. of a braid group ψ_n:B_n→End(V^⊗n)

quantum inv. of 3-manifolds quantum rep. of a mapping class group

ρ:M1,1 →End( ˆA( ))

· · ·

· · ·

· · ·

σ

σ∈B_n,L= ˆσ Jones poly.

trace(h^⊗n·ψn(σ))

T

T ∈M_1,1,M =S³_ˆ

T

Casson inv.

?

Aim B

Construct the representation of the mapping class group MCG(Σ1,1) and present the Casson-Walker invariant ofg= 1 open books

Open book decompositions

Anopen book decompositions (genus 1, 1 boundary comp) of a 3-manifoldM

M = ^Σ1,1×[0,1] ^ϕ _∪

D²×S¹

whereφ: a homeo ofΣ1,1 s.t. φ|∂Σ1,1 = id_∂Σ1,1

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Surgery presentation

Fact : A 3-mfd with genus one open book decomp has the following surgery presentation.

Ln1,···,n_N;m

•

•

•

•

•

n_N

n3

n₂ n₁

n_N−1

mhalf twists

half twist

S³ M

n₁,···,n_N;m surgery alongLⁿ1,···,n_N;m

Result A

M_{n1,···,nN;m} : 3-mfd with genus one open book decomp obtained by surgery along a linkL_{n1,···,nN;m} Suppose thatMn1,···,nN;m is a QHS,

Theorem A [M.]

λ(M_{n1,···,nN;m}) =− 1 24(∑

i

n_i−3σ)

−(−1)^m+σ+ 24|H1| (2∑

i

ni+ 6N−12m) whereσ : signature of linking matrix ofLn1,···,nN;m

σ+: ♯ posi. eigenvalues of linking mat.

|H₁|: order ofH₁(M_{n1,···,nN;m};Z)

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Result B

M_φ : 3-mfd with genus one open book decomp which monodromy is φ

Suppose thatMφ is a QHS

˜

φ∈M]1,1 =T2/KI^′, KII, KIII ρ:M]_1,1→End( ˆA( )) Theorem B [M.]

The Casson-Walker invariant ofMφ can be calculated as follows.

λ(Mφ) = 2

tr0(ρ( ˜φ))tr1(ρ( ˜φ)) +1 8σ( ˜φ)

whereσ( ˜φ) : signature of linking matrix ofLφ˜

tri(ρ( ˜φ)) =degree ipart of “quantum trace” of ρ( ˜φ)

A table of contents

Introduction

• Backgrounds

• Motivations

• Def (Open book decomp) and Prop (Surgery presentation)

• Results Part A

• Def (Surgery) and Explanation (Surgery presentation)

• Idea of Proof of Thm A

• Definitions (Jacobi diag, Kontsevich inv, LMO inv)

• Proof of Thm A Part B

• Definitions (Kirby moves, Space of Jacobi diags, Product)

• Construction of Representation

• Proof of Thm B

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Surgery

S³ M

the surgery alongL S³

L

pull out the tubular nbd ofL

ψ

D²×S¹

M = (S³\the nbd ofL)∪ψ (⊔^ℓD²×S¹)

Surgery presentations and monodromies

⇝

α β

⇝

↔α

⇝

↔β

•

•

•

⇝

•

•

•

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Idea of Proof of Thm A

λ(M)=

deg=1 LMO inv Z₁^LMO(M)

Kontsevich inv of L Z(L)

ι

Caluculate

glue them

Z

Z(L

n₁,···,n_N;m) Z^LMO

1 (M

n₁,···,n_N;m)

ι

Jacobi diagrams

AJacobi diag on a 1-manifoldX

degof a Jacobi diag = ¹₂ #vertices

A(X) = span_C {Jacobi diags onX}/AS, IHX, STU rel

AS : = -

IHX : = -

STU : = -

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Kontsevich invariant

TheKontsevich invis an inv of a link L

Z(L) =Z(T1)◦Z(T2)◦ · · · ◦Z(Tk)∈ A(⊔^ℓS¹) (T_i : elementary q-tangle)

For example,

Z( ) = Φ = + 1

24[ , ] +· · ·

Z( ) =ν¹² =





 ^S2Φ )







−¹2

Z( ) = ^exp ₂ = +1

2 +1

8 + 1

48 +· · ·

deg=1 part of LMO invariant

Thedeg=1 part of LMO inv Z₁^{LM O}(M) = ι(Z(L))

ι(Z( ))^σ+ι(Z( ))^σ− ∈span_C{∅, } ι:A(⊔^ℓS¹)→span_C{∅, }

where

ι : 7−→

7−→ 1 2 7−→ 1

6 + 1

6

=−2

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The value of the deg=1 LMO invariant of g = 1 open books

Mn1,···,nN;m : 3-mfd with genus one open book decomp obtained by surgery along a linkL_{n1,···,nN;m} A_N : linking matrix ofL_{n1,···,nN;m}

SupposedetAN ̸= 0

Z₁^LMO(M) =c₀(M) +c₁(M)θ Prop

c₁(M_{n1,···,nN;m}) =− 1

48(−1)^N+σ+detA_N(trA_N −3σ)

− 1

48(−1)^m+N+σ+(2trA_N+ 6N −12m)

The value of a clasp

Z( ) =

ν¹2

ν¹2

S1Φ

S₁S₃∆2Φ

S2exp

−

S₁S₃Φ

= ^exp − 1

24 + 1

96 + 1

96 (Ohtsuki, 2007)

where ^exp = +1

2 +1

8 + 1

48 +· · ·

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Result A

M_{n1,···,nN;m} : 3-mfd with genus one open book decomp obtained by surgery along a linkL_{n1,···,nN;m} Suppose thatMn1,···,nN;m is a QHS,

Theorem A [M.]

λ(M_{n1,···,nN;m}) =− 1 24(∑

i

n_i−3σ)

−(−1)^m+σ+ 24|H1| (2∑

i

ni+ 6N−12m) whereσ : signature of linking matrix ofLn1,···,nN;m

σ+: ♯ posi. eigenvalues of linking mat.

|H₁|: order ofH₁(M_{n1,···,nN;m};Z)

Proof of Thm A

Fact : λ(M) = 2c1(M)

|H₁| when b1(M) = 0 Jacobi diags which become throughι

, , , , , ,

, ,

As for non circular case, it can be calculated as LMO inv of lens spaces.

As for circular case, we have only to claculate above three diags. 2

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The value of twist knots 1

n

m Kn;m

M_n;m=S_K³

n;m, |H₁(M_n;m)|=|n+ (−1)^m2|

c1(S_K³_n,m) =













(−1)^σ+(₁

48(n+ 2)(n−3σ) + ₄₈¹(2n+ 6 + 12m)) m : even (−1)^σ+(₁

48(n−2)(n−3σ)− ₄₈¹(2n+ 6 + 12m)) m : odd

λ(Mn;m) =

{−₂₄^{1 ((n+2)−1)((n+2)}_n+2 ⁻²⁾ −_2(n+2)^m (m:even, n+ 2>0)

−₂₄^{1 ((n−2)−1)((n}_n−2 ⁻²⁾⁻²⁾ −_2(n^m+1₋₂₎ (m:odd, n−2>0)

The value of twist knots 2

λ(M_n;m) =

{−₂₄^{1 ((n+2)−1)((n+2)}_n+2 ⁻²⁾ −_2(n+2)^m (m:even, n+ 2>0)

−₂₄^{1 ((n−2)−1)((n}_n−2 ⁻²⁾⁻²⁾ −_2(n^m+1₋₂₎ (m:odd, n−2>0)

λ(S_K³) =λ(S³) + 1

2p∆^′′_K(1)−s(1, p)

=−s(1, p) + 1

2p∆^′′_K(1) s(1, p) = (p−1)(p−2)

12p

∆^′′_Kn,m(1) =

{−m m : even m+ 1 m : odd

22 / 34

A table of contents

Introduction

• Backgrounds

• Motivations

• Def (Open book decomp) and Prop (Surgery presentation)

• Results Part A

• Def (Surgery) and Explanation (Surgery presentation)

• Idea of Proof of Thm A

• Definitions (Jacobi diag, Kontsevich inv, LMO inv)

• Proof of Thm A Part B

• Definitions (Kirby moves, Space of Jacobi diags, Product)

• Construction of Representation

• Proof of Thm B

Kirby moves

the KI move : ←→ ∅ ←→

the KII move : −→

the KIII move :

0

−→ ∅

theKI^′ move : ←→ ∅

24 / 34

The space of the Jacobi diagrams on 2-tangles

Aˆ( ) ={Jacobi diagrams on 2-tangles up to AS, IHX, STU}/P₂, O₁, I_>2

P₂ : + + ∼0

O1 : ∼ −2

I_>2 : the Jacobi diagram whose ♯trivalent vertices >2∼0 We set 10 elements as the basis ofA( ).

µ₀₀= ,µ₁ = ,µ₁₀= ,µ₀₁= ,µ₁₁= , θµ₀₀= ⊔ ,θµ₁ = ⊔ ,θµ₁₀= ⊔ , θµ₀₁= ⊔ ,θµ₁₁= ⊔

The product on the space of the Jacobi diagrams

The product on the spaceA( )

•: ˆA( )⊗A(ˆ )→^◦ A(Sˆ ¹⊔ )→^ˆι A(ˆ )

For diagramsη= ^D ,η^′= ^D′ ∈ A( ),

η•η^′= ˆι







 _D^′

D







∈A(ˆ ),

26 / 34

Construction of Representation

ρ:M]1,1=T2/KI^′, KII, KIII→^ˆιZˇ Aˆ( )→^π End( ˆA( )) ˆ

ι:=√

−1^ℓι: ˆA((⊔^ℓS¹)⊔ )→A(ˆ ) ˆιZˇ is invariat under the KI^′, KII, KIII moves.

•: ˆA( )⊗Aˆ( )∋ (

D , ^D′ )

7→ˆι





D^′ D



∈Aˆ( )

π: ˆA( )→End( ˆA( )) π(

ˆ ιZ(R)ˇ )

: ˆA( )→A(ˆ ) ˆ

ιZˇ(T)7→ˆιZˇ(R◦T) = ˆιZ(R)ˇ •ˆιZ(R)ˇ

Result B

M_φ : 3-mfd with genus one open book decomp which monodromy is φ

Suppose thatM_φ is a QHS

˜

φ∈M]1,1 =T2/KI^′, KII, KIII ρ:M]_1,1→End( ˆA( )) Theorem B [M.]

The Casson-Walker invariant ofM_φ can be calculated as follows.

λ(Mφ) = 2

tv0ρ( ˜φ)w

tv1ρ( ˜φ)w+1 8σ( ˜φ) whereσ( ˜φ) : signature of linking matrix ofL_φ˜ v_i =degree ipart of^t

(

tr ,tr ,tr ,tr ,tr ,· · ·) w= ˆιZ(ˇ )∈Aˆ( )

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Proof of Thm B

It is known that

Z₁^LMO(M) =c0(M) +c1(M)θ, λ(M) = 2

c₀(M)c1(M), Z₁^LMO(M) =

(

−1 + 1 16θ

)_−σ+( 1 + 1

16θ )_−σ−

ιZ(L)ˇ

=(−1)^σ+ (

1 + 1 16σθ

)

(b0+b1θ)

=(−1)^σ+b₀+ (−1)^σ+ (σ

16b₀+b₁ )

θ.

On the other hand, we have that

(trµ₀₀,trµ₁,trµ₁₀,· · ·)ρ( ˜φ)ˆιZ(ˇ ) = ˆιZ(Lˇ _φ˜).

2

Concrete Calculations

When the monodromyφis periodic in M1,1, we can set

˜

φ=h^mαⁿ¹β· · ·α^nNβ, ∀i, ni ≤0, ∃j, nj ̸= 0,

whereh= ◦ ,α=

+1 ,β = ⁺¹ ,

andσ( ˜φ) = 0.

ρ(h),ρ(α),ρ(β) can be presented as matrices inGL10(C).

v=^t(0, −2, −2, −2, 1

6θ, 0, −2θ, −2θ, −2θ, 0) w=^t(0, −1, 1

2, 1 2, 0, 1

24, 0, 0, 0, 0) Thus, the Casson-Walker invaritant ofMφ is

λ(M_φ) = 2

tv0ρ( ˜φ)w

tv₁ρ( ˜φ)w

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Presentation of the generators

α= 1

24θµ₀₀+ (

−1 + 1 24

) µ₁+1

2µ₁₀+1

2µ₀₁+1 4

( 1− 1

3θ )

µ₁₁

β= (

−1 + 1 16θ

)

µ00−µ1+1

2(1− 1

48θ)µ10+1

2(1− 1

48θ)µ01− 1 96θµ11

V =V₁⊕V₂⊕V₃

=C⟨µ₀₀, µ₁₀, θµ₀₀, θµ₁₀⟩ ⊕C⟨µ₀₁, µ₁₁, θµ₀₁, θµ₁₁⟩ ⊕C⟨µ₁, θµ₁⟩

Concrete matrix presentaion of α

A:=ρ(

+1

) is the following matrix.













1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

1

2 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ¹₂ 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ₂₄¹ 0 0 0 0 1 0 0 0

1

96 0 ₂₄¹ 0 0 ¹₂ 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 −₉₆¹ −₂₄¹ 0 0 0 ¹₂ 1













=

( _{1 0 0 0}

1

2 1 0 0

0 0 1 0

−₉₆¹ −₂₄^{1 1}₂ 1

)

⊕

( _{1 0 0 0}

1

2 1 0 0

0 0 1 0

−₉₆¹ −₂₄^{1 1}₂ 1

)

⊕( _{1 0}

−₂₄¹ 1

)

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Concrete matrix presentaion of β

B:=ρ( ⁺¹ ) is the following matrix.













1 0 −2 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0

0 0 0 1 −2 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0

−₄₈¹ 0 ₂₄¹ 0 0 1 0−2 0 0 0 −₂₄¹ 0 0 0 0 1 0 0 0 0 0 −₄₈¹ 0 0 0 0 1 0 0 0 0 0 −48¹

1

24 0 0 0 1−2 0 0 0 0 −₄₈¹ 0 0 0 0 1













=

( _{1 −2 0 0}

0 1 0 0

−₄₈¹ ₂₄¹ 1−2 0 −₄₈¹ 0 1

)

⊕

( _{1 −2 0 0}

0 1 0 0

−₄₈¹ ₂₄¹ 1−2 0 −₄₈¹ 0 1

)

⊕( _{1 0}

−₂₄¹ 1

)

Summary and Future directions

Summary

• We calculated the Casson-Walker invariant ofg= 1 open books through the calculation of the deg=1 part of the LMO invariant.

• We constructed the representaion of M_1,1 on the space of the Jacobi diagrams on , and we gave a formula for the calculation of the Casson-Walker invariant of g= 1 open books.

Future directions

• relation to rep. theory of MCG

• relation to contact topology

• general cases

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