On non-acyclic SL 2 -representations of twist knots

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Introduction Non-acyclic rep’s (−3)-Dehn-surgery L-functions

On non-acyclic SL 2 -representations of twist knots

植木潤（

Jun Ueki

^）

,

^{東京電機大学（}

Tokyo Denki University

^）

joint work with Ryoto Tange and Anh T. Tran.

December 18, 2019

結び目の数理 II, 日本大学文理学部百周年記念館 B. Mazur’s problem.

Investigate the L-function of the universal deformation ρ over Z

p

= lim ←− Z/p

ⁿ

Z of a non-acyclic residual rep’n ρ : π → SL

2

( F

p

).

Previous observation [KMTT2018]:

We have L = (x ˙ − α)

²

in R

univ

= Z

p

[[x − α]] for several examples.

We study certain irreducible SL

2

( C )-representations of twist knot groups.

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1

Introduction Twist knot Theorems A&B

2

Non-acyclic rep’s

Character variety f

n

(x , y ) = 0 and Reidemeister torsion τ

n

(x , y ) Proof of Theorem A

3

(−3)-Dehn-surgery Proof of Theorem B

4

L-functions

Universal deformation

Twisted Alexander polynomials

M

²

KR-dictionary

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Twist knot Theorems A&B

Twist knot

For each n ∈ Z , the twist knot J(2, 2n) is define by the diagram below, the horizontal twists being right handed if n > 0 and left handed if n < 0.

-full twists n

We have J(2, 0) = 0

1

(unknot), J(2, 2) = 3

1

(trefoil), J(2, 4) = 5

2

, and J(2, − 2) = 4

1

(figure-eight knot).

The knot group π

n

:= π

1

(S

³

− J(2, 2n)) admits the following presentation:

π

n

= ⟨a, b | aw

ⁿ

= w

ⁿ

b⟩ , w = [a, b

⁻¹

] = ab

⁻¹

a

⁻¹

b.

Conjugacy classes of ρ : π

n

→ SL

2

( C ) are parametrized by x := tr ρ(a) and y := tr ρ(ab).

We also use z := tr ρ(w ) = 2x

²

− x

²

y + y

²

− 2.

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Twist knot Theorems A&B

Theorems A&B

We consider irreducible SL

2

(C)-rep’s ρ of π

n

= π

1

(S

³

− J(2, 2n)), whose conj.

classes are parametrized by x = tr ρ(a) and y = tr ρ(b).

Theorem (A)

The set of all conj. classes of non-acyclic ρ’s are exactly given by x = y = 1 − 2 cos 2πk

3n − 1 , 0 < k ≤ 3n − 1 2 . ρ : π

n

→ SL

2

( C ) is acyclic ⇐⇒

def

H

i

(S

³

− J(2, 2n), ρ) = 0 for all i . Theorem (B)

Irreducible SL

2

( C )-representations of J(2, 2n) on x = y are exactly those which factor through the (−3)-Dehn surgery.

We also obtain the common tangent property, which yields L = (x ˙ − α)

²

.

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Character varietyfn(x,y) = 0and Reidemeister torsionτn(x,y) Proof of Theorem A

Chebyshev polynomials

For each n ∈ Z , we define the Chebyshev polynomials ς

n

(z) ∈ Z [z] by ς

n

(2 cos θ) = sin nθ

sin θ ,

or equivalently, ς

0

(z) = 0, ς

1

(z) = 1, and ς

n+1

(z) − zς

n

(z) + ς

n−1

(z) = 0.

Then we have ς

_±1

(z) = ± 1, ς

_±2

(z) = ± z, ς

_±3

(z) = ± (z

²

− 1), . . ..

For every n ∈ Z , we have ς

n

( ± 2) = ( ± 1)

ⁿ⁻¹

n.

If z ̸ = ± 2, then we may write z = s + s

⁻¹

, and we have ς

n

(z) = (s

ⁿ

+ s

⁻ⁿ

)/(s + s

⁻¹

).

These equalities yield ς

n+1

(z)

²

+ ς

n

(z)

²

− zς

n+1

(z)ς

n

(z) = 1.

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Character variety f n (x, y) = 0 and Reidemeister torsion τ n (x, y )

Now recall z := tr ρ(w ) = 2x

²

− x

²

y + y

²

− 2.

Proposition ([Le1993]+[NagasatoTran2016])

Put f

n

(x, y ) := (y − 1)ς

n

(z ) − ς

n−1

(z ) ∈ Z[x, y]. Then the SL

2

(C)-character variety of π

n

is given by (x

²

− y − 2)f

n

(x , y ) = 0.

Each class of irreducible rep’s corresponds to each point on f

n

(x, y) = 0 with x

²

− y − 2 ̸= 0.

Proposition ([Tran2016])

The Reidemeister torsions of acyclic rep’s extends to a polynomial function τ

n

= τ

n

(x , y ) ∈ Z [x , y ] on the curve f

n

(x , y ) = 0.

We have τ

n

(x , y ) = 0 ⇐⇒

iﬀ

ρ at (x , y ) is non-acyclic.

If n moves, then we have τ

n

= − 2(z − x )ς

n

(z) + (x − 2)(ς

_n−1

+ 2)

z − 2 , or

equivalently, τ

0

= 0, τ

1

= −2, and τ

n+1

− zτ

n

+ τ

n−1

− 2(x − 2) = 0.

Non-acyclic rep’s correspond to the intersection {f

n

(x, y) = 0} ∩ {τ

n

(x, y ) = 0}.

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f n (x, y ) = 0 and τ n (x, y) = 0 for n = − 3, − 2, − 1, 2, 3, and the line y = x

-4 -2 0 2 4

Theorem (C)

The curves f

n

(x, y ) = 0 and τ

n

(x , y ) = 0 in R

²

have a common tangent line at

their every intersection point.

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f n (x, y ) = 0 and τ n (x, y) = 0 for n = − 3, − 2, − 1, 2, 3, and the line y = x

-4 -2 0 2 4

Theorem (C)

The curves f

n

(x, y ) = 0 and τ

n

(x , y ) = 0 in R

²

have a common tangent line at

their every intersection point.

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Proof of Theorem A

f

n

= τ

n

= 0 = ⇒ x = y : We have (y − 1)ς

n

(z) = ς

n−1

(z) · · · ⃝,

¹

(z − x )ς

n

+ (x − 2)(y − 1)ς

n

+ (x − 2) = 0 · · · ⃝

²

, and

ς

n+1

(z)

²

+ ς

n

(z)

²

− zς

n+1

(z)ς

n

(z ) = 1 · · · ( ∗ ).

These equalities yield (y − 2)(x − y )

²

(x

²

− y − 2) = 0.

We have x

²

− y − 2 ̸= 0 by the assumption. We may verify y − 2 ̸= 0.

Hence we have x − y = 0.

Common roots of f

n

(x , x) and τ

n

(x , x) :

We may verify z = s + s

⁻¹

̸ = ± 2 and write x − 1 = − (α + α

⁻¹

) for α ∈ C

^∗

. Then we have z = −x

³

+ 3x

²

− 2 = · · · = α

³

+ α

⁻³

,

ς

n

(z) =

^s_sⁿ₋^−s_s₋₁⁻ⁿ

=

^α_α³ⁿ₃^−α₋_α⁻³ⁿ₋₃

, and ς

_n−1

(z ) =

^sⁿ⁻¹_s^−s₋_s⁻⁽ⁿ⁻¹⁾₋₁

=

^α³⁽ⁿ⁻¹⁾_α₃^−α₋_α^−3(n−1)₋₃

. By ς

n

(z) =

_x(x¹₋₂₎

and ς

n−1

(z) =

_x_(x^x⁻₋¹₂₎

, we obtain

(α

³ⁿ⁺¹

+ 1)(α

³ⁿ⁻¹

− 1) = 0 and (α

³ⁿ⁻¹

− 1)(α

³ⁿ⁻⁵

+ 1) = 0.

We easily see α

³ⁿ⁻¹

= 1, hence α = e

ⁱ³ⁿ⁻¹^2πk

for 0 < k < |3n − 1|,

hence x = 1 − (α + α

⁻¹

) = 1 − 2 cos

_3n^2πk₋₁

for 0 < k ≤

^|³ⁿ⁻₂¹^|

.

(Note that x ̸ = − 1, 0, 2.)

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Proof of Theorem B

Proof of Theorem B

Recall π

n

= ⟨ a, b | aw

ⁿ

= w

ⁿ

b ⟩ , w = [a, b

⁻¹

] = ab

⁻¹

a

⁻¹

b.

If we take µ = a as a meridian, then its preferred longitude is λ = w

ⁿ

w

ⁿ

. The result of ( − 3)-Dehn surgery is π

1

(M

₋₃

) = π

n

/ ⟨

µ

⁻³

λ ⟩

= π

n

/ ⟨

a

⁻³

w

ⁿ

w

ⁿ

⟩ . We have Riley’s representation given by

ρ(a) =

( M 1 0 M

⁻¹

)

, ρ(b) =

( M 0

− u M

⁻¹

)

, where − u = y − x

²

+ 2 for x = tr ρ(a) = M + M

⁻¹

and y = tr ρ(ab).

We in addition have ρ(λ) = ( L c

0 L

⁻¹

)

for some L and c.

Since µ and λ commutes, ρ(µ) and ρ(λ) are simultaneously diagonalizable, and we see that ρ factors through π

1

(M

₋3

) ⇐⇒

iﬀ

L = M

³

holds.

We have u = (1 − M

²

)(1 − L)

L + M

²

· · · (⋆) by [HosteShanahan2004, (5,9)].

Proof of Theorem B : If L = M

³

, then by noting M + 1 ̸ = 0, (⋆) yields x = y .

If x = y , then by noting M

⁴

̸= ±1, (⋆) yields L = M

³

.

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Proof of Theorem B

Appendix. A remark on ( − 3)-Dehn surgery

The twist knot J(2, 2n) is hyperbolic iﬀ n ̸= 0, 1.

The ( − 3)-Dehn surgery for a hyperbolic twist knot is an exceptional surgery.

Indeed, M

₋3

is Seifert fibered by [BrittenhamWu2001, Theorem 1.1].

If we instead consider J(−2, 2n), then the similar assertion to Theorem B holds

for the 3-Dehn surgery.

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Universal deformation Twisted Alexander polynomials M2KR-dictionary

Universal deformation

For a residual representation ρ : π

n

→ SL

2

( F

p

), the universal deformation ρ : π

n

→ SL

2

(R

ρ

) (def’d up to certain conjugation) is the universal object to all lifts ρ : π

n

→ SL

2

(A) of ρ to complete local Z

p

-algebras.

We usually obtain the universal deformation of ρ by localizing Riley’s universal representation

ρ(a) = (

x+

√

4−x²

2

1

0

^x⁻

√

4−x² 2

)

, ρ(b) = (

x+

√

4−x²

2

0 y − x

²

+ 2

^x⁻

√

4−x² 2

)

at a lift of ρ, so that we have R

ρ

= Z

p

[[x − α]] for some α ∈ Z

p

.

We define the twisted Alexander polynomials ∆

ρ,i

(t) to be the generators of

FittH g

i

(π

n

, ρ). The L-function of ρ is defined to be L

ρ

= ∆ ˙

ρ,1

(1).

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L-functions

We define the twisted Alexander polynomials ∆

ρ,i

(t) to be the generators of FittH g

i

(π

n

, ρ). The L-function of ρ is defined to be L

ρ

= ∆ ˙

ρ,1

(1).

By [HillmanLivingstonNaik2006, Theorem 11] together with a well-known calculation of Reidemeister torsion with use of the Fox derivative, we have Proposition

L

ρ

= ˙ τ

ρ

· ∆

ρ,0

(1) in Z

p

[[x − α]].

Since Z

p

[[x − α]] is a ring of dimension 2, we have something involved and Lemma (KMTT2018, Proposition 3.8.1)

If L

ρ

̸ = 1, then ˙ ρ is non-acyclic.

If ρ is non-acyclic, then we have x − 2 ˙ ̸= 1 in Z

p

[[x − α]], hence ∆

ρ,0

(1) ˙ = 1.

Therefore L

ρ

is calculated as the localization τ

ρ

of τ

ρ

for Riley’s ρ at a non-acyclic representation ρ.

The common tangent property (Theorem C) yields L

ρ

= (x ˙ − α)

²

.

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M ² KR-dictionary

The common tangent property (Theorem C) yields L

ρ

= (x ˙ − α)

²

. However, the geometry behind this property (and “x = y ”) is yet to be revealed.

This work is in the scope of the following dictionary (cf. [Morishita2012, Chapter 14]).

Low dimensional topology Number theory Deformation space Universal p-ordinary of hyperbolic structures modular deformation space Dehn surgery points with Z -coeﬀ. Arithmetic points References:

[KMTT2018] Takahiro Kitayama, Masanori Morishita, Ryoto Tange, and Yuji Terashima, On certain L-functions for deformations of knot group representations, Trans. Amer. Math. Soc. 370 (2018), 3171–3195.

[Morishita2012] Masanori Morishita, Knots and primes, Universitext, Springer, London, 2012, An introduction to arithmetic topology.

[TTU] Ryoto Tange, Anh T. Tran, and Jun Ueki, preprint.

Thank you for your attention!!

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M ² KR-dictionary

The common tangent property (Theorem C) yields L

ρ

= (x ˙ − α)

²

. However, the geometry behind this property (and “x = y ”) is yet to be revealed.

This work is in the scope of the following dictionary (cf. [Morishita2012, Chapter 14]).

Low dimensional topology Number theory Deformation space Universal p-ordinary of hyperbolic structures modular deformation space Dehn surgery points with Z -coeﬀ. Arithmetic points References:

[KMTT2018] Takahiro Kitayama, Masanori Morishita, Ryoto Tange, and Yuji Terashima, On certain L-functions for deformations of knot group representations, Trans. Amer. Math. Soc. 370 (2018), 3171–3195.

On non-acyclic SL 2 -representations of twist knots