[PPT] - On the twisted Alexander polynomial for metabelian SL 2 ( C ) PowerPoint Presentation

SLIDE 1

On the twisted Alexander polynomial for metabelian 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.

SLIDE 2

Basic notion and notation

The twisted Alexander polynomial The twisted Alexander polynomial = 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 → ρ(γ) → Adρ(γ) : v → ρ(γ)vρ(γ)−1 sl2(C) = C 0 1

0 0

⊕ C

1 0

0 −1

⊕ C

0 0

1 0

The adjoint action Ad gives a connection with the character

variety Hom(π1(EK ), SL2(C))//SL2(C).

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

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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.

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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 ∗ a−1

a ∈ C \ {0}
⊂ SL2(C)

ρ : irreducible

Def

⇐ ⇒ ρ : not reducible.

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

SLIDE 5

Reducible and irreducible reps. in metabelian ones

ρ : π1(EK ) → SL2(C) metabelian ρ : reducible ρ : [π1(EK ), π1(EK )] →

±

1 ω 1

ω ∈ C
⊂ SL2(C)

since Im ρ ⊂ a ∗ a−1

a ∈ C \ {0}
and

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

ρ : irreducible

ρ : [π1(EK ), π1(EK )] → a a−1

a ∈ C \ {0}
⊂ SL2(C)

ρ(µ) = 1 −1

by F. Nagasato.

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

SLIDE 6

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.

SLIDE 7

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◦ρ

EK

(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.

SLIDE 8

Hierarchy of metabelian representations

Remark ρ : π1(EK ) → SL2(C) reducible = ⇒ ρ : metabelian (∵) Im ρ ⊂ a

∗ 0 a−1

a = 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.

SLIDE 9

The details of reducible metabelian representations

Existence of reducible representations (G. Burde, G. de Rham) ∃ρ : π1(EK ) → SL2(C) : reducible and non–abelian if 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.

SLIDE 10

The twisted Alexander polynomial of reducible SL2(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◦ρ

EK

(t) = ∆K(λ2t) (λ2t − 1) · ∆K (t) (t − 1) · ∆K(λ−2t) (λ−2t − 1).

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

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The details of irreducible metabelian representations

Existence of irreducible metabelian reps. (Nagasato) ♯{the characters of irereducible metabelian reps.} = |∆K (−1)| − 1 2 . ι ∈ H1(EK ; Z2) induces an involution ˆ ι on Hom(π1(EK ), SL2(C))//SL2(C). Remark (Nagasato & Y.) {the characters of irereducible metabelian reps.} = {the fixed point set of ˆ ι} 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.

SLIDE 12

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◦ρ

EK

(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.

SLIDE 13

The details of ∆α⊗Ad◦ρ

EK

(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 )]) = {1}. α : π1(EK ) → π1(EK)/[π1(EK ), π1(EK )] ≃ H1(EK ) = t

s.t. α(µ) = t

the twisted Alexander poly. ∆α⊗Ad◦ρ

EK

(t) = det

α ⊗ Ad ◦ ρ
∂ri

∂gj

i, j=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.

SLIDE 14

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.

SLIDE 15

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.

SLIDE 16

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.

SLIDE 17

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.

SLIDE 18

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 in S3 \ 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.

SLIDE 19

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 in S3 \ 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.

SLIDE 20

Remark & Examples of Lin’s presentation

Remark The 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−1

2 x1µ−1 = x−1 2

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.

SLIDE 21

Explicit form of metabelian reps.

X-S. Lin, F. Nagasato The correspondence µ → 1 −1

,

xi → ξi ξ−1

i

gives a metabelian rep.

They gives all representatives of conj. classes of metabelian reps. ♯ (conj. classes of irred. metabelian reps) = |∆K (−1)| − 1 2 .

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

SLIDE 22

Sketch of proof

Ad ◦ ρ : π1(EK ) → Aut(sl2(C)) = Aut(C 0 1

0 0

⊕ C

1 0

0 −1

⊕ C

0 0

1 0

)

µ →   −1 −1 −1   xi →   ξ2 1 ξ−2   The subspace C 1 0

0 −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.

SLIDE 23

Observation about ∆α⊗Ad◦ρ

EK

(t)

∆α⊗Ad◦ρ

EK

(t) = ∆α⊗ρ2

EK

(t) · ∆α⊗ρ1

EK

(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.

SLIDE 24

Details in the proof

The representations ρ1 and ρ2 are given by ρ1 : µ → −1, xi → 1, ρ2 : µ → −1 −1

,

xi → ξ2

i

ξ−2

i

.

Wada’s criterion ψ : π1(EK ) → GLn(R), R : UFD If ∃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 → ξ2

i

ξ−2

i

(xi ∈ [π1(EK), π1(EK )])

Wada’s criterion − − − − − − − − − − − − − → ∆α⊗ρ2

EK

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

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

SLIDE 25

Decomposition of Q(t)

ρ : longitude-regular ⇒ ∆α⊗Ad◦ρ

EK

(1) = 0, ∆α⊗Ad◦ρ

EK

(1) = Q(1) · ∆K(−1) −2 = 0 ⇒ Q(1) = 0. Set C = √ −1 − √ −1

,

Cρ2C−1 : µ → 1 1

= −ρ(µ), xi →

ξ2

i

1 1 ξ−2

i

= ρ(xi)
inv. of conj.

− − − − − − − − − → Q(t) = Q(−t). Therefore Q(t) = (t − 1)(t + 1)P(t)

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

SLIDE 26

Observation about ∆α⊗Ad◦ρ

EK

(t) (again)

∆α⊗Ad◦ρ

EK

(t) = ∆α⊗ρ2

EK

(t) · ∆α⊗ρ1

EK

(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.

SLIDE 27

Examples

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

2

= 1).

∃1 irred. metabelian rep. (up to conjugate) Lin presentation π1(EK ) =

µ, x1, x2
µx1µ−1 = x1x−1

2 ,

µx−1

2 x1µ−1 = x−1 2

ρ(µ) =
1

−1

, ρ(x1) =

ζ3 ζ−1

3

, ρ(x2) =

ζ2

3

ζ−2

3

where ζ3 = e

2π√−1 3

⇒ ∆α⊗Ad◦ρ

EK

(t) = (t − 1)∆K(−t).

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

SLIDE 28

figure eight knot

Figure: a free Seifert surface of the figure eight knot

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

SLIDE 29

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

2

= 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) =

1 −1

, ρ(x1) =

ζ5 ζ−1

5

, ρ(x2) =

ζ2

5

ζ−2

5

(2) ρ(µ) =
1

−1

, ρ(x1) =

ζ2

5

ζ−2

5

, ρ(x2) =

ζ4

5

ζ−4

5

where ζ5 = e

2π√−1 5

⇒ ∆α⊗Ad◦ρ

EK

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

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

SLIDE 30

52 knot

Figure: a free Seifert surface of 52 knot

Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.

SLIDE 31

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

2

= 3).

∃3 irred. metabelian rep. (up to conjugate) Lin presentation π1(EK ) =

µ, x1, x2
µx1µ−1 = x1x−1

2 ,

µx−2

2 x1µ−1 = x−2 2

(1) ρ(µ) =

1 −1

, ρ(x1) =

ζ7 ζ−1

7

, ρ(x2) =

ζ2

7

ζ−2

7

(2) ρ(µ) =

1 −1

, ρ(x1) =

ζ2

7

ζ−2

7

, ρ(x2) =

ζ4

7

ζ−4

7

(3) ρ(µ) =

1 −1

, ρ(x1) =

ζ3

7

ζ−3

7

, ρ(x2) =

ζ6

7

ζ−6

7

where ζ7 = e

2π √ −1 7 Yoshikazu Yamaguchi The twisted Alexander poly. for metabelian reps.