PDF Alexander-Oka Polynomials Twisted Alexander Polynomials

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Alexander-Oka Polynomials Twisted Alexander Polynomials

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Recent Progress on Topology of Plane Curves: A Quick Trip

Part IV:

Other Generalizations of Alexander Polynomials

Twisted and Alexander-Oka polynomials

José Ignacio COGOLLUDO-AGUSTÍN

Departamento de Matemáticas Universidad de Zaragoza

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Contents

1 Alexander-Oka Polynomials

Settings and Definitions Examples

Remarks and Open Problems

2 Twisted Alexander Polynomials

Franz-Reidemeister Torsion of a Complex Fox calculus

Example

Remarks, Questions and Open Problems

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Settings

C=C₁∪...∪ C_r ⊂P²

G:=π₁(P²\ C₀∪ C).

ε: H1(P²\ C₀∪ C) →→ Z

γ_i 7→ ε_i

Definition (Oka,−, Libgober)

TheAlexander polynomial∆C,ε(t)ofCwith respect toεis the order of Kε/K_ε⁰ as aQ[t^±1]-module.

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Settings

C=C₁∪...∪ C_r ⊂P²

G:=π₁(P²\ C₀∪ C).

ε: H1(P²\ C₀∪ C) →→ Z

γ_i 7→ ε_i

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Settings

C=C₁∪...∪ C_r ⊂P²

G:=π₁(P²\ C₀∪ C).

ε: H1(P²\ C₀∪ C) →→ Z

γ_i 7→ ε_i

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Settings

C=C₁∪...∪ C_r ⊂P²

G:=π₁(P²\ C₀∪ C).

ε: H1(P²\ C₀∪ C) →→ Z

γ_i 7→ ε_i

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Properties

Remarks

Oka, Randell.∆C,ε(t)coincides with the characteristic polynomial of the monodromy of the Milnor fiber ofF := ΠF_i^εⁱ.

−, Libgober.

∆_C,ε(t)| Y

P∈Sing(C)

∆_C,ε,P_i(t)Y

i

(t^εⁱ−1)^kⁱ,

∆C,ε(t)|∆C,ε,C₀(t).

Artal,−, Tokunaga.Consider the evaluation morphism

Q[t₁^±1, ...,t_r^±1] →^ϕ^ε Q[t^±1] t_i 7→ t^εⁱ then∆˜_C,ε(t)is the generator ofϕ˜_ε(F1(C)).

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Properties

Remarks

−, Libgober.

∆_C,ε(t)| Y

P∈Sing(C)

∆_C,ε,P_i(t)Y

i

(t^εⁱ−1)^kⁱ,

∆C,ε(t)|∆C,ε,C₀(t).

Artal,−, Tokunaga.Consider the evaluation morphism

Q[t₁^±1, ...,t_r^±1] →^ϕ^ε Q[t^±1] t_i 7→ t^εⁱ then∆˜_C,ε(t)is the generator ofϕ˜_ε(F1(C)).

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Properties

Remarks

−, Libgober.

∆_C,ε(t)| Y

P∈Sing(C)

∆_C,ε,P_i(t)Y

i

(t^εⁱ−1)^kⁱ,

∆C,ε(t)|∆C,ε,C₀(t).

Artal,−, Tokunaga.Consider the evaluation morphism Q[t^±1, ...,t_r^±1] →^ϕ^ε Q[t^±1]

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Alexander Polynomials of a curve

Theorem (Libgober,-)

The Alexander polynomial ofCw.r.t.εis the first invariant of the colored Burau representation matrix of the braid monodromy ofC w.r.t.εdivided by(1−t^P^εⁱ)/(1−t).

Colored Burau Representation:

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Alexander Polynomials of a curve

σ₁7→







−t^εⁱ 1 0 ... 0

0 1 0 ... 0

0 0 1 ... 0

... ...

0 0 0 ... 1







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Alexander Polynomials of a curve

σ_i 7→







1 ... 0 ...

0 ... 1

0 0

0

1 0 0

t^εⁱ −t^εⁱ 1

0 0 1

0

0 0

1 ... 0 ...

0 ... 1







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Examples

Consider example from Part I:

∆_C₁(t) = (t−1).

∆_C₂(t) = (t−1).

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Examples

Consider example from Part I:

∆_C₁(t) = (t−1).

∆_C₂(t) = (t−1).

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Examples

However, ifε:= (1,2), then

∆_C₁_,ε(t) = (t−1).

∆_C₂_,ε(t) =(t²−1).

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Examples

However, ifε:= (1,2), then

∆_C₁_,ε(t) = (t−1).

∆_C₂_,ε(t) =(t²−1).

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Examples

Moreover, ifε:= (2,1), then

∆_C₁_,ε(t) = (t−1).

∆_C₂_,ε(t) =(t−1)(t²+1).

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Examples

Moreover, ifε:= (2,1), then

∆_C₁_,ε(t) = (t−1).

∆_C₂_,ε(t) =(t−1)(t²+1).

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Remarks and Open Problems

Remarks

From only some∆_C,ε(t)can be determined bygeometrical conditions on the singular points ofC.

Are there geometrical conditions for each∆_C,ε(t)?

Find a (maybe universal) bound on∆_C,ε(t)in terms of degC. The roots of∆_C,ε(t)ared-th roots of unity (d :=degF).

Find a better (maybe universal) bound on the multiplicities of the roots of∆_C,ε(t).

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Remarks and Open Problems

Remarks

From only some∆_C,ε(t)can be determined bygeometrical conditions on the singular points ofC.

Are there geometrical conditions for each∆_C,ε(t)?

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Remarks and Open Problems

Remarks

From only some∆_C,ε(t)can be determined bygeometrical conditions on the singular points ofC. Are there geometrical conditions for each∆_C,ε(t)?

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Remarks and Open Problems

Remarks

Find a (maybe universal) bound on∆C,ε(t)in terms of degC.

The roots of∆_C,ε(t)ared-th roots of unity (d :=degF).

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Remarks and Open Problems

Remarks

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Remarks and Open Problems

Remarks

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Example

Franz-Reidemeister Torsion of a Complex

1 C∗a finite chain complex:

C∗=Cn

−→ · · ·∂ −→^∂ C0,

2 C_i finite dimensionalF-vector spaces, and∂◦∂=0.

3 Choose a basisc_i forC_i,

4 Chooseh¯i a basis for the homologyHi(C∗),

5 Chooseh_i a lift of¯h_i inC_i.

6 Choosebi a basis of the image of∂:Ci+1−→Ci and

7 Chooseeb_i a lift ofb_i inC_i+1.

8 b_ih_ieb_i₋₁is a basis ofC_i.

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Example

Franz-Reidemeister Torsion of a Complex

C∗=Cn

−→ · · ·∂ −→^∂ C0,

3 Choose a basisc_i forC_i,

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Example

Franz-Reidemeister Torsion of a Complex

C∗=Cn

−→ · · ·∂ −→^∂ C0,

3 Choose a basisci forCi,

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Example

Franz-Reidemeister Torsion of a Complex

C∗=Cn

−→ · · ·∂ −→^∂ C0,

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Example

Franz-Reidemeister Torsion of a Complex

C∗=Cn

−→ · · ·∂ −→^∂ C0,

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Example

Franz-Reidemeister Torsion of a Complex

C∗=Cn

−→ · · ·∂ −→^∂ C0,

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Example

Franz-Reidemeister Torsion of a Complex

C∗=Cn

−→ · · ·∂ −→^∂ C0,

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Example

Franz-Reidemeister Torsion of a Complex

C∗=Cn

−→ · · ·∂ −→^∂ C0,

8 bihiebi−1is a basis ofCi.

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Example

Definition

TheFranz-Reidemeister torsionof(C∗;c,h)is

τ(C∗;c,h) :=

n

Y

i=0

[bihiebi−1|c_i]⁽⁻¹⁾ⁱ⁺¹∈F^∗/{±1}.

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Example

Consider a finitely presented groupG=hx1, . . . ,xm|r1, . . . ,rni.

ρ:G→GLr(F[t^±1])a representation ofG. ε:H1(G)→Za surjective homomorphism. There is a ring homomorphism

Z[G] −→ GL_r(F[t^±1]) γ 7−→ t^ε(γ)ρ(γ).

LetFmbe the free group generated byx1, . . . ,xm. Set

Φ :Z[F_m]−→Z[G]−→^ε⊗ρ GL_r(F[t^±1]).

There exists somei such thatΦ(x_i−1)has a non-zero determinant. Letp_i : (λ^r)^m−→(λ^r)^m−1be the projection in the direction of thei-th copy ofλ^r.

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Example

ρ:G→GLr(F[t^±1])a representation ofG.

ε:H1(G)→Za surjective homomorphism. There is a ring homomorphism

Z[G] −→ GL_r(F[t^±1]) γ 7−→ t^ε(γ)ρ(γ).

Φ :Z[F_m]−→Z[G]−→^ε⊗ρ GL_r(F[t^±1]).

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Example

ε:H1(G)→Za surjective homomorphism.

There is a ring homomorphism

Z[G] −→ GL_r(F[t^±1]) γ 7−→ t^ε(γ)ρ(γ).

Φ :Z[F_m]−→Z[G]−→^ε⊗ρ GL_r(F[t^±1]).

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Example

Z[G] −→ GL_r(F[t^±1]) γ 7−→ t^ε(γ)ρ(γ).

Φ :Z[F_m]−→Z[G]−→^ε⊗ρ GL_r(F[t^±1]).

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Example

Z[G] −→ GL_r(F[t^±1]) γ 7−→ t^ε(γ)ρ(γ).

Φ :Z[F_m]−→Z[G]−→^ε⊗ρ GL_r(F[t^±1]).

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Example

Z[G] −→ GL_r(F[t^±1]) γ 7−→ t^ε(γ)ρ(γ).

Φ :Z[F_m]−→Z[G]−→^ε⊗ρ GL_r(F[t^±1]).

There exists somei such thatΦ(x_i−1)has a non-zero determinant.

Letp_i : (λ^r)^m−→(λ^r)^m−1be the projection in the direction of thei-th

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Example

Z[G] −→ GL_r(F[t^±1]) γ 7−→ t^ε(γ)ρ(γ).

Φ :Z[F_m]−→Z[G]−→^ε⊗ρ GL_r(F[t^±1]).

There exists somei such thatΦ(x_i−1)has a non-zero determinant.

Letp_i : (λ^r)^m−→(λ^r)^m−1be the projection in the direction of thei-th copy ofλ^r.

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Example

Consider the(nr×mr)-matrix

Υ =h Φ ∂r_k

∂xl

i ,

Qi :=

gcd{ r(m−1)×r(m−1)

-minors of(piΥ)} ifn≥m

1 otherwise

One can define thetwisted Alexander polynomialof(π;ε, ρ)as

∆_X,ε,ρ(t) :=Qi/det(Φ(xi−1)).

Theorem (Kirk, Livingston, Wada)

Let X be a finite CW-complex. If H₁^ε,ρ(X;F[t^±1])is torsion, then τε,ρ(X) = ∆_X,ε,ρ(t).

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Example

Υ =h Φ ∂r_k

∂xl

i ,

Qi :=

gcd{ r(m−1)×r(m−1)

1 otherwise

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Example

Υ =h Φ ∂r_k

∂xl

i ,

Qi :=

gcd{ r(m−1)×r(m−1)

1 otherwise

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Example

Υ =h Φ ∂r_k

∂xl

i ,

Qi :=

gcd{ r(m−1)×r(m−1)

1 otherwise

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Example

1 G1:=B3=hx1,x2:r1≡x1x2x1(x2x1x2)⁻¹i

2 ρ(x₁) :=

1 0

−1 1

ρ(x₂) := 1 1

0 1

3 ε(x₁) =ε(x₂) =1.

4

∂r1

∂x₁ =∂x1

∂x₁ +ρ(x1)t ∂x2x1(x2x1x2)⁻¹

∂x₁ =

1 0 0 1

+

1 0

−1 1

t ∂x₂

∂x1

+ρ(x₂)t ∂x₁(x₂x₁x₂)⁻¹

∂x1

=

...

t²−t+1 t(t−1)

−t² 1−t

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Example

1 G1:=B3=hx1,x2:r1≡x1x2x1(x2x1x2)⁻¹i

2 ρ(x₁) :=

1 0

−1 1

ρ(x₂) :=

1 1 0 1

3 ε(x₁) =ε(x₂) =1.

4

∂r1

∂x₁ =∂x1

∂x₁ +ρ(x1)t ∂x2x1(x2x1x2)⁻¹

∂x₁ =

1 0 0 1

+

1 0

−1 1

t ∂x₂

∂x1

+ρ(x₂)t ∂x₁(x₂x₁x₂)⁻¹

∂x1

=

...

t²−t+1 t(t−1)

−t² 1−t

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Example

1 G1:=B3=hx1,x2:r1≡x1x2x1(x2x1x2)⁻¹i

2 ρ(x₁) :=

1 0

−1 1

ρ(x₂) :=

1 1 0 1

3 ε(x₁) =ε(x₂) =1.

4

∂r1

∂x₁ =∂x1

∂x₁ +ρ(x1)t ∂x2x1(x2x1x2)⁻¹

∂x₁ =

1 0 0 1

+

1 0

−1 1

t ∂x₂

∂x1

+ρ(x₂)t ∂x₁(x₂x₁x₂)⁻¹

∂x1

=

...

t²−t+1 t(t−1)

−t² 1−t

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Example

1 G1:=B3=hx1,x2:r1≡x1x2x1(x2x1x2)⁻¹i

2 ρ(x₁) :=

1 0

−1 1

ρ(x₂) :=

1 1 0 1

3 ε(x₁) =ε(x₂) =1.

4

∂r1

∂x₁ =

∂x1

∂x₁ +ρ(x1)t ∂x2x1(x2x1x2)⁻¹

∂x₁ =

1 0 0 1

+

1 0

−1 1

t ∂x₂

∂x1

+ρ(x₂)t ∂x₁(x₂x₁x₂)⁻¹

∂x1

=

...

t²−t+1 t(t−1)

−t² 1−t

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Example

1 G1:=B3=hx1,x2:r1≡x1x2x1(x2x1x2)⁻¹i

2 ρ(x₁) :=

1 0

−1 1

ρ(x₂) :=

1 1 0 1

3 ε(x₁) =ε(x₂) =1.

4

∂r1

∂x₁ =∂x1

∂x₁ +ρ(x₁)t ∂x2x1(x2x1x2)⁻¹

∂x₁ =

1 0 0 1

+

1 0

−1 1

t ∂x₂

∂x1

+ρ(x₂)t ∂x₁(x₂x₁x₂)⁻¹

∂x1

=

...

t²−t+1 t(t−1)

−t² 1−t

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Example

1 G1:=B3=hx1,x2:r1≡x1x2x1(x2x1x2)⁻¹i

2 ρ(x₁) :=

1 0

−1 1

ρ(x₂) :=

1 1 0 1

3 ε(x₁) =ε(x₂) =1.

4

∂r1

∂x₁ =∂x1

∂x₁ +ρ(x₁)t ∂x2x1(x2x1x2)⁻¹

∂x₁ =

1 0 0 1

+

1 0

−1 1

t ∂x₂

∂x1

+ρ(x₂)t ∂x₁(x₂x₁x₂)⁻¹

∂x1

=

...

t²−t+1 t(t−1)

−t² 1−t

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Example

1 G1:=B3=hx1,x2:r1≡x1x2x1(x2x1x2)⁻¹i

2 ρ(x₁) :=

1 0

−1 1

ρ(x₂) :=

1 1 0 1

3 ε(x₁) =ε(x₂) =1.

4

∂r1

∂x₁ =∂x1

∂x₁ +ρ(x₁)t ∂x2x1(x2x1x2)⁻¹

∂x₁ =

1 0 0 1

+

1 0

−1 1

t ∂x₂

∂x1

+ρ(x₂)t ∂x₁(x₂x₁x₂)⁻¹

∂x1

=

...

t²−t+1 t(t−1)

−t² 1−t

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Example

1 G1:=B3=hx1,x2:r1≡x1x2x1(x2x1x2)⁻¹i

2 ρ(x₁) :=

1 0

−1 1

ρ(x₂) :=

1 1 0 1

3 ε(x₁) =ε(x₂) =1.

4

∂r1

∂x₁ =∂x1

∂x₁ +ρ(x₁)t ∂x2x1(x2x1x2)⁻¹

∂x₁ =

1 0 0 1

+

1 0

−1 1

t ∂x₂

∂x1

+ρ(x₂)t ∂x₁(x₂x₁x₂)⁻¹

∂x1

=

...

t²−t+1 t(t−1)

−t² 1−t

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Example

Φ(∂r₁

∂xi

) =

t²−t+1 t(t−1) t−1 −t²

−t² 1−t t²−t −t²+t−1

SinceΦ(x₂−1) = 1 1

0 1

t− 1 0

0 1

=

t−1 0

−t t−1

and

detΦ(x₂−1) = (t−1)²

one obtains

∆_G₁_,ε,ρ(t) :=

t²−t+1 t(t−1)

−t² 1−t

(t−1)² = (t²+1).

Note that∆_G₁(t) =t²−t+1 for the classical Alexander polynomial.

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Example

Φ(∂r₁

∂xi

) =

t²−t+1 t(t−1) t−1 −t²

−t² 1−t t²−t −t²+t−1

SinceΦ(x₂−1) = 1 1

0 1

t− 1 0

0 1

=

t−1 0

−t t−1

and

detΦ(x₂−1) = (t−1)²

one obtains

∆_G₁_,ε,ρ(t) :=

t²−t+1 t(t−1)

−t² 1−t

(t−1)² = (t²+1).

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Example

Φ(∂r₁

∂xi

) =

t²−t+1 t(t−1) t−1 −t²

−t² 1−t t²−t −t²+t−1

SinceΦ(x₂−1) = 1 1

0 1

t− 1 0

0 1

=

t−1 0

−t t−1

and

detΦ(x₂−1) = (t−1)²

one obtains

∆_G₁_,ε,ρ(t) :=

t²−t+1 t(t−1)

−t² 1−t

(t−1)² = (t²+1).

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Example

Φ(∂r₁

∂xi

) =

t²−t+1 t(t−1) t−1 −t²

−t² 1−t t²−t −t²+t−1

SinceΦ(x₂−1) = 1 1

0 1

t− 1 0

0 1

=

t−1 0

−t t−1

and

detΦ(x₂−1) = (t−1)²

one obtains

∆_G₁_,ε,ρ(t) :=

t²−t+1 t(t−1)

−t² 1−t

(t−1)² = (t²+1).

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Example

Φ(∂r₁

∂xi

) =

t²−t+1 t(t−1) t−1 −t²

−t² 1−t t²−t −t²+t−1

SinceΦ(x₂−1) = 1 1

0 1

t− 1 0

0 1

=

t−1 0

−t t−1

and

detΦ(x₂−1) = (t−1)²

one obtains

∆_G₁_,ε,ρ(t) :=

t²−t+1 t(t−1)

−t² 1−t

(t−1)² = (t²+1).

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Example

G2:=B4=hx1,x2,x3:r1≡x1x2x1(x2x1x2)⁻¹, r2≡ x1x2x1(x2x1x2)⁻¹, r3≡x1x3(x3x1)⁻¹i

ρ(x1) =ρ(x3) :=

1 0

−1 1

ρ(x2) := 1 1

0 1

ε(x₁) =ε(x₂) =ε(x₃) =1.







t²−t+1 t(t−1) t−1 −t² 0 0

−t² 1−t t²−t −t²+t−1 0 0

1−t 0 0 0 t−1 0

t 1−t 0 0 −t t−1

0 0 1−t t² −t²+t−1 t−t²

0 0 −t(t−1) t²−t+1 t² t−1







detΦ(x₃−1) = (t−1)²

∆_G₂_,ε,ρ(t) = (t−1)(t²+1).

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Example

ρ(x1) =ρ(x3) :=

1 0

−1 1

ρ(x2) :=

1 1 0 1

ε(x₁) =ε(x₂) =ε(x₃) =1.







t²−t+1 t(t−1) t−1 −t² 0 0

−t² 1−t t²−t −t²+t−1 0 0

1−t 0 0 0 t−1 0

t 1−t 0 0 −t t−1

0 0 1−t t² −t²+t−1 t−t²

0 0 −t(t−1) t²−t+1 t² t−1







detΦ(x₃−1) = (t−1)²

∆_G₂_,ε,ρ(t) = (t−1)(t²+1).

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Example

ρ(x1) =ρ(x3) :=

1 0

−1 1

ρ(x2) :=

1 1 0 1

ε(x1) =ε(x2) =ε(x3) =1.







t²−t+1 t(t−1) t−1 −t² 0 0

−t² 1−t t²−t −t²+t−1 0 0

1−t 0 0 0 t−1 0

t 1−t 0 0 −t t−1

0 0 1−t t² −t²+t−1 t−t²

0 0 −t(t−1) t²−t+1 t² t−1







detΦ(x₃−1) = (t−1)²

∆_G₂_,ε,ρ(t) = (t−1)(t²+1).

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Example

G2:=B