on the twisted alexander polynomial for metabelian sl2(c)-...

On the twisted Alexander polynomial formetabelian SL2(C)–representations

with the adjoint action

Yoshikazu Yamaguchi

JSPS Research fellow (PD)Tokyo Institute of Technology

RIMS Seminar“Twisted topological invariants and topology of

low-dimensional manifolds”

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Basic notion and notation

The twisted Alexander polynomial

The twistedAlexanderpolynomial

=A refinement of ∆K (t)(the Alexander polynomial)with ρ : π1 → GL(V )

Notation

EK := S3 \ N(K ) a knot exterior,

Ad ◦ ρ : π1(EK )ρ−→ SL2(C)

Ad−−→ Aut(sl2(C))

γ 7→ ρ(γ) 7→ Adρ(γ) : v 7→ ρ(γ)vρ(γ)−1

sl2(C) = C(

0 10 0

)

⊕ C(

1 00 −1

)

⊕ C(

0 01 0

)

The adjoint action Ad gives a connection with the charactervariety Hom(π1(EK ),SL2(C))//SL2(C).

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Metabelian representations

Definition of metabelian reps.

ρ : π1(EK ) → SL2(C) is metabelian

⇐⇒ ρ([π1(EK ), π1(EK )]) ⊂ SL2(C) an abelian subgroup.

Remark

ρ : π1(EK ) → SL2(C) is abelian⇐⇒ ρ(π1(EK )) ⊂ SL2(C) an abelian subgroup,

⇐⇒ ρ([π1(EK ), π1(EK )]) = {1},

⇐⇒π1(EK ) SL2(C)

H1(EK ; Z) π1(EK )/[π1(EK ), π1(EK )] ≃ 〈µ〉

ρ

ρ is determined by only ρ(µ).

We focus on non–abelian representations.

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Definitions of reducible and irreducible reps.

ρ : π1(EK ) → SL2(C)

ρ : reducible

Def⇐⇒ ∃L ⊂ C2 s.t. ρ(g)(L) ⊂ L (∀g ∈ π1(EK ))

by taking conjugation,

ρ : π1(EK ) →{(

a ∗0 a−1

)∣

∣

∣

∣

a ∈ C \ {0}}

⊂ SL2(C)

ρ : irreducible

Def⇐⇒ ρ : not reducible.

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Reducible and irreducible reps. in metabelian ones

ρ : π1(EK ) → SL2(C) metabelianρ : reducible

ρ : [π1(EK ), π1(EK )] →{

±(

1 ω0 1

)∣

∣

∣

∣

ω ∈ C

}

⊂ SL2(C)

since Im ρ ⊂{(

a ∗0 a−1

)∣

∣

∣

∣

a ∈ C \ {0}}

and

A,B ∈ SL2(C) : upper triangular ⇒ ABA−1B−1 =

(

1 ∗0 1

)

ρ : irreducible

ρ : [π1(EK ), π1(EK )] →{(

a 00 a−1

)∣

∣

∣

∣

a ∈ C \ {0}}

⊂ SL2(C)

ρ(µ) =

(

0 1−1 0

)

by F. Nagasato.

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Background

ρ : π1(EK ) → SL2(C) metabelian

ρ : reducible∆K (t) appears in the twisted Alexander polynomial.

−→ Hyperbolic torsion at “bifurcation points”in Hom(π1(EK ,SL2(C)))//SL2(C).

ρ : irreducible

“Does ∆K (t) appear in the twisted Alexander?”

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Main Theorem

Theorem

Suppose that

ρ : π1(EK ) → SL2(C) s.t.{

an irred. metabelian and;“longitude–regular ”.

Then(the twisted Alexander poly.)

∆α⊗Ad◦ρEK

(t) ·= (t − 1)∆K (−t)P(t),

where ∆K (t) is the Alexander polynomial of K .

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Hierarchy of metabelian representations

Remark

ρ : π1(EK ) → SL2(C) reducible =⇒ ρ : metabelian

(∵) Im ρ ⊂{( a ∗

0 a−1

) ∣

∣ a 6= 0}

⇒ ρ([π1(EK ), π1(EK )]) ⊂{

±(

1 ∗0 1

)}

We have the following hierarchy of metabelian representations:

abelian ⊂ reducible ⊂ metabelian

roots of ∆K (t) ∆K (−1)

difference difference

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

The details of reducible metabelian representations

Existence of reducible representations (G. Burde, G. de Rham)

∃ρ : π1(EK ) → SL2(C) : reducible and non–abelianif and only if∆K (λ2) = 0 where λ is an eigenvalue of ρ(µ).

{the characters of non–abelian reducible representations}= Cabel ∩ Cnon–abel in Hom(π1(EK ),SL2(C))//SL2(C),

where

Cabel is the component of abelian characters and

Cnon–abel is the components of non–abelian ones.

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

The twisted Alexander polynomial of reducibleSL2(C)-representations

Theorem

ρ : π1(EK ) → SL2(C) reducible and non–abelian,

corresponding to λ ∈ C s.t.{

λ is an eigenvalue of ρ(µ) and∆K (λ2) = 0

,

then

∆α⊗Ad◦ρEK

(t) =∆K (λ2t)(λ2t − 1)

· ∆K (t)(t − 1)

· ∆K (λ−2t)(λ−2t − 1)

.

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

The details of irreducible metabelian representations

Existence of irreducible metabelian reps. (Nagasato)

♯{the characters of irereducible metabelian reps.}=

|∆K (−1)| − 12

.

ι ∈ H1(EK ; Z2) induces

an involution ι̂ on Hom(π1(EK ),SL2(C))//SL2(C).

Remark (Nagasato & Y.)

{the characters of irereducible metabelian reps.}= {the fixed point set of ι̂} in Hom(π1(EK ),SL2(C))//SL2(C).

Remark

A higher rank analogy was given by H. Boden and S. Friedl.

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Main Theorem (again)

Theorem

Suppose that

ρ : π1(EK ) → SL2(C) s.t.{

an irred. metabelian and;“longitude–regular ”.

Then(the twisted Alexander poly.)

∆α⊗Ad◦ρEK

(t) ·= (t − 1)∆K (−t)P(t),

where ∆K (t) is the Alexander polynomial of K .

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

The details of ∆α⊗Ad◦ρEK

(t)

Homomorphisms

ρ : π1(EK ) → SL2(C) is metabelian

⇐⇒ ρ([π1(EK ), π1(EK )])(⊂ SL2(C)) is abelian,

Suppose that ρ([π1(EK ), π1(EK )]) 6= {1}.α : π1(EK ) → π1(EK )/[π1(EK ), π1(EK )] ≃ H1(EK ) = 〈t〉s.t. α(µ) = t

the twisted Alexander poly.

∆α⊗Ad◦ρEK

(t) =

det

(

α⊗ Ad ◦ ρ(

∂ri∂gj

)

i , j 6=1

)

det (α⊗ Ad ◦ ρ (g1 − 1))

from a presentation π1(EK ) = 〈g1,g2, . . . ,gk | r1, . . . , rk−1〉.

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Main tools

We need a “good” presentation of π1(EK )for metabelian reps.X-S. Lin introduced a suitable presentation of π1(EK )by using a free Seifert surface of K .a Seifert surface S is free⇔ S3 = S × [−1,1] ∪ S3 \ S × [−1,1]

: a Heegaard splitting.

Figure: a free Seifert surface of the trefoil knot

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Main tools

We need a “good” presentation of π1(EK )for metabelian reps.X-S. Lin introduced a suitable presentation of π1(EK )by using a free Seifert surface of K .a Seifert surface S is free⇔ S3 = S × [−1,1] ∪ S3 \ S × [−1,1]

: a Heegaard splitting.

Figure: a free Seifert surface of the trefoil knot

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Main tools

We need a “good” presentation of π1(EK )for metabelian reps.X-S. Lin introduced a suitable presentation of π1(EK )by using a free Seifert surface of K .a Seifert surface S is free⇔ S3 = S × [−1,1] ∪ S3 \ S × [−1,1]

: a Heegaard splitting.

Figure: a free Seifert surface of the trefoil knot

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Main tools

We need a “good” presentation of π1(EK )for metabelian reps.X-S. Lin introduced a suitable presentation of π1(EK )by using a free Seifert surface of K .a Seifert surface S is free⇔ S3 = S × [−1,1] ∪ S3 \ S × [−1,1]

: a Heegaard splitting.

Figure: a free Seifert surface of the trefoil knot

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Lin’s presentation

By using a free Seifert surface S with genus 2g,

π1(EK ) = 〈µ, x1, . . . , x2g |µa+i µ

−1 = a−i (i = 1, . . . 2g)〉

where

xi is a closed loop corresponding to 1–handle inS3 \ S × [−1,1],

a±i is a word in x1, . . . , x2g , corresponding to closed loops

in the spine of S.

Figure: a free Seifert surface of the trefoil knot

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Lin’s presentation

By using a free Seifert surface S with genus 2g,

π1(EK ) = 〈µ, x1, . . . , x2g |µa+i µ

−1 = a−i (i = 1, . . . 2g)〉

where

xi is a closed loop corresponding to 1–handle inS3 \ S × [−1,1],

a±i is a word in x1, . . . , x2g , corresponding to closed loops

in the spine of S.

Figure: a free Seifert surface of the trefoil knot

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Remark & Examples of Lin’s presentation

RemarkThe generators x1, . . . , x2g are null–homologous,

i.e., xi ∈ [π1(EK ), π1(EK )].

K = trefoil knot

π1(EK ) =

⟨

µ, x1, x2

∣

∣

∣

∣

µx1µ−1 = x1x−1

2 ,

µx−12 x1µ

−1 = x−12

⟩

K = figure eight knot

π1(EK ) =

⟨

µ, x1, x2

∣

∣

∣

∣

µx1µ−1 = x1x−1

2 ,

µx2x1µ−1 = x2

⟩

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Explicit form of metabelian reps.

X-S. Lin, F. Nagasato

The correspondence

µ 7→(

0 1−1 0

)

, xi 7→(

ξi 00 ξ−1

i

)

gives a metabelian rep.

They gives all representatives of conj. classes ofmetabelian reps.

♯ (conj. classes of irred. metabelian reps) =|∆K (−1)| − 1

2.

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Sketch of proof

Ad ◦ ρ : π1(EK ) → Aut(sl2(C))

= Aut(C(

0 10 0

)

⊕ C(

1 00 −1

)

⊕ C(

0 01 0

)

)

µ 7→

−1−1

−1

xi 7→

ξ2

1ξ−2

The subspace C(

1 00 −1

)

is invariant space.

Decomposition of Ad ◦ ρ

Ad ◦ ρ = ρ2 ⊕ ρ1

where ρ1 is 1–dim. rep. and ρ2 is 2–dim. rep.

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Observation about ∆α⊗Ad◦ρEK

(t)

∆α⊗Ad◦ρEK

(t) = ∆α⊗ρ2EK

(t) · ∆α⊗ρ1EK

(t)

= Q(t) · ∆K (−t)(−t − 1)

Wada Milnor

= (t − 1)(t + 1)P(t) · ∆K (−t)−t − 1

longitude–regular& inv. of conj.

for R–torsion

= −(t − 1) · P(t) · ∆K (−t).

Remark

The P(t) satisfies that P(t) = P(−t).

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Details in the proof

The representations ρ1 and ρ2 are given by

ρ1 : µ 7→ −1, xi 7→ 1,

ρ2 : µ 7→(

0 −1−1 0

)

, xi 7→(

ξ2i 00 ξ−2

i

)

.

Wada’s criterion

ψ : π1(EK ) → GLn(R), R : UFDIf ∃γ ∈ [π1(EK ), π1(EK )] s. t. 1 is not an eigenvalue of ψ(γ),⇒ ∆α⊗ψ

EK(t) is a Laurent polynomial.

ρ2 : xi 7→(

ξ2i 00 ξ−2

i

)

(xi ∈ [π1(EK ), π1(EK )])

Wada’s criterion−−−−−−−−−−−−−→ ∆α⊗ρ2EK

(t) = Q(t) (Laurent polynomial)

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Decomposition of Q(t)

ρ : longitude-regular ⇒ ∆α⊗Ad◦ρEK

(1) = 0,

∆α⊗Ad◦ρEK

(1) = Q(1) · ∆K (−1)

−2= 0 ⇒ Q(1) = 0.

Set C =

(√−1 00 −

√−1

)

,

Cρ2C−1 : µ 7→(

0 11 0

)

= −ρ(µ), xi 7→(

ξ2i 11 ξ−2

i

)

= ρ(xi)

inv. of conj.−−−−−−−−−→ Q(t) = Q(−t).

ThereforeQ(t) = (t − 1)(t + 1)P(t)

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Observation about ∆α⊗Ad◦ρEK

(t) (again)

∆α⊗Ad◦ρEK

(t) = ∆α⊗ρ2EK

(t) · ∆α⊗ρ1EK

(t)

= Q(t) · ∆K (−t)(−t − 1)

Wada Milnor

= (t − 1)(t + 1)P(t) · ∆K (−t)−t − 1

longitude–regular& inv. of conj.

for R–torsion

= −(t − 1) · P(t) · ∆K (−t).

Remark

The P(t) satisfies that P(t) = P(−t).

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

Examples

trefoil knot ∆K (t) = t2 − t + 1 ( |∆K (−1)|−12 = 1).

∃1 irred. metabelian rep. (up to conjugate)Lin presentation

π1(EK ) =

⟨

µ, x1, x2

∣

∣

∣

∣

µx1µ−1 = x1x−1

2 ,

µx−12 x1µ

−1 = x−12

⟩

ρ(µ) =

(

0 1−1 0

)

, ρ(x1) =

(

ζ3 00 ζ−1

3

)

, ρ(x2) =

(

ζ23 00 ζ−2

3

)

where ζ3 = e2π

√

−13

⇒ ∆α⊗Ad◦ρEK

(t) = (t − 1)∆K (−t).

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

figure eight knot

Figure: a free Seifert surface of the figure eight knot

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

figure eight knot ∆K (t) = t2 − 3t + 1 ( |∆K (−1)|−12 = 2).

∃2 irred. metabelian reps. (up to conjugate)Lin presentation

π1(EK ) =

⟨

µ, x1, x2

∣

∣

∣

∣

µx1µ−1 = x1x−1

2 ,µx2x1µ

−1 = x2

⟩

(1) ρ(µ) =

(

0 1−1 0

)

, ρ(x1) =

(

ζ5 00 ζ−1

5

)

, ρ(x2) =

(

ζ25 00 ζ−2

5

)

(2) ρ(µ) =

(

0 1−1 0

)

, ρ(x1) =

(

ζ25 00 ζ−2

5

)

, ρ(x2) =

(

ζ45 00 ζ−4

5

)

where ζ5 = e2π

√

−15

⇒ ∆α⊗Ad◦ρEK

(t) = (t − 1)∆K (−t).(the both reps. have the same result)

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

52 knot

Figure: a free Seifert surface of 52 knot

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

52 knot ∆K (t) = 2t2 − 3t + 2 ( |∆K (−1)|−12 = 3).

∃3 irred. metabelian rep. (up to conjugate)Lin presentation

π1(EK ) =

⟨

µ, x1, x2

∣

∣

∣

∣

µx1µ−1 = x1x−1

2 ,

µx−22 x1µ

−1 = x−22

⟩

(1) ρ(µ) =

(

0 1−1 0

)

, ρ(x1) =

(

ζ7 00 ζ−1

7

)

, ρ(x2) =

(

ζ27 00 ζ−2

7

)

(2) ρ(µ) =

(

0 1−1 0

)

, ρ(x1) =

(

ζ27 00 ζ−2

7

)

, ρ(x2) =

(

ζ47 00 ζ−4

7

)

(3) ρ(µ) =

(

0 1−1 0

)

, ρ(x1) =

(

ζ37 00 ζ−3

7

)

, ρ(x2) =

(

ζ67 00 ζ−6

7

)

where ζ7 = e2π

√

−17

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

(1) ρ(µ) =

(

0 1−1 0

)

, ρ(x1) =

(

ζ7 00 ζ−1

7

)

, ρ(x2) =

(

ζ27 00 ζ−2

7

)

⇒ ∆α⊗Ad◦ρEK

(t) = (t−1)(2+e6π√−1/7+e−6π

√−1/7)∆K (−t),

(2) ρ(µ) =

(

0 1−1 0

)

, ρ(x1) =

(

ζ27 00 ζ−2

7

)

, ρ(x2) =

(

ζ47 00 ζ−4

7

)

⇒ ∆α⊗Ad◦ρEK

(t) = (t−1)(2+e2π√−1/7+e−2π

√−1/7)∆K (−t),

(3) ρ(µ) =

(

0 1−1 0

)

, ρ(x1) =

(

ζ37 00 ζ−3

7

)

, ρ(x2) =

(

ζ67 00 ζ−6

7

)

⇒ ∆α⊗Ad◦ρEK

(t) = (t−1)(2+e4π√−1/7+e−4π

√−1/7)∆K (−t).

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.