Automorphisms of surfaces and Seifert ﬁbered surgeries on ﬁbered knots

(1)

Automorphisms of surfaces and Seifert ﬁbered surgeries

on ﬁbered knots

Kimihiko Motegi (Nihon Univ.) joint work with

Kazuhiro Ichihara (Osaka Sangyo Univ.)

(2)

Exceptional surgeries on hyperbolic, ﬁbered knots

K : ﬁbered knot in a closed 3-manifold M .

E (K ) = ˆ F × [0, 1]/(x, 0) = ( ˆ f (x), 1);

mapping torus of a once punctured, com- pact, orientable surface ˆ F via a monodromy map ˆ f : ˆ F → F ˆ .

F

E(K)

^ f

^

(3)

Dehn ﬁllings and

capping oﬀ monodromies

An existence of

a Seifert ﬁbered (resp. toroidal) surgery on a hyperbolic, ﬁbered knot K

⇐⇒

An existence of

a pseudo-Anosov monodromy of ˆ F

whose capped oﬀ monodromy is isotopic to a periodic (resp. reducible) automorphism.

[Thurston], [Otal], [Jaco]

2

(4)

F f

E(K)

f ^

^

(K;0) Dehn filling

x

0

capping of f

hyperbolic toroidal

Seifert fibered p s eudo-Anosov reducible

periodic F

(5)

Longitudinal, toroidal surgery

There is a pseudo-Anosov monodromy of a hyperbolic, ﬁbered knot K in S ³

whose capped oﬀ monodromy is isotopic to a reducible automorphism,

i.e., (K; 0) is a toroidal manifold. [Gabai]

We can ﬁnd inﬁnitely many such phenomena by Osoinach’s construction.

4

(6)

What can we say about longitudinal, Seifert ﬁbered surgery?

There is no known example of a pseudo-Anosov monodromy of a hyperbolic, ﬁbered knot K in S ³

whose capped oﬀ monodromy is isotopic to a periodic automorphism,

i.e., (K; 0) is a Seifert ﬁber space.

In fact, Teragaito conjectures that there are

“no such examples”.

(7)

If the monodromy ˆ f has a prong ≥ 2 singular- ity at the boundary, then the invariant mea- sured singular foliation on ˆ F can be naturally extended to that of the capped oﬀ surface F . [suggestion by J.Los]

capping off

prong = 3 prong = 3

F F

^

Thus we have:

6

(8)

Proposition 1 Let K be a hyperbolic, fibered knot in a closed 3-manifold

with a monodromy isotopic to a pseudo-Anosov automorphism having a prong n ≥ 2 singu- larity at the boundary.

Then the resulting manifold obtained by Dehn surgery along the fiber slope is hyperbolic.

In particular, it is not a Seifert fiber space.

(9)

Examples in homology 3-spheres

For any integer g ≥ 2,

there is a homology 3-sphere M and a hyperbolic, fibered knot K of genus g in M such that a longitudinal surgery on K yields a Seifert ﬁber space.

In other words,

There is a pseudo-Anosov monodromy of a hyperbolic, ﬁbered knot in homology-S ³ whose capped oﬀ monodromy is isotopic to a periodic automorphism.

8

(10)

Construction

T _p,q : (p, q)-torus knot in S ³

Assume q > p ≥ 2 and (p, q) = (2, 3).

Then (T _p,q ; 0) = (F × I )/ { (x, 0) = (f (x), 1) } ,

where F is a closed surface of genus ⁽ ^p ⁻ ¹⁾⁽ ₂ ^q ⁻ ¹⁾ and f is a periodic monodromy of period pq ﬁxing a single point x ₀ ∈ F .

M _f = (T _p,q ; 0) is a Seifert ﬁber space.

Choose a hyperbolic section s of M _f passing (x ₀ , 0) = (x ₀ , 1).

x₀ F

F [0, 1]

f

x₀

s

(11)

Attach a solid torus V to M _f − intN (s) to ob- tain a homology 3-sphere Σ and a hyperbolic, ﬁbered knot K ⊂ Σ which is a core of V .

Dually, a surgery on K ⊂ Σ along the ﬁber slope yields a Seifert ﬁber space M _f .

For a given integer g ≥ 2,

by starting with (2, 2g + 1)-torus knot in the above,

we have the required examples.

(12)

The above construction is based on

an existence of a hyperbolic section in M _f , i.e., a section with hyperbolic complement.

It should be noted that if (p, q) = (2, 3), then there is no such a section in M _f = (T ₂ _, ₃ ; 0).

The construction leads us:

Question

How can we describe the position of

hyperbolic sections in a surface bundle over the circle?

(13)

Visualizing sections on a surface

“3-dimensional view → 2-dimensional view”

f t

x

₀

F

F [0, 1]

F : orientable, closed surface of genus ≥ 2.

f : automorphism of F ﬁxing a speciﬁed point x ₀ ∈ F

t : monotone arc in F × [0, 1] connecting (x ₀ , 0) and (x ₀ , 1)

11

(14)

s = t / f

x

₀

F

projection of s

f

M f

c:

M _f = F × [0, 1]/(x, 0) = (f (x), 1) : mapping

torus which is a surface bundle over S ¹ Then t deﬁnes a section s ⊂ M _f .

The projection c deﬁnes an element [c] ∈ π ₁ (F, x ₀ ).

[c] = [c ] ∈ π ₁ (F, x ₀ ) ⇒ s _c and s _c

are isotopic.

Question

Can we describe hyperbolic sections by their “projections” on the surface F ?

(15)

Theorem 2 Suppose that the monodromy f is irreducible and periodic with period p.

s _c is hyperbolic.

⇐⇒

[c]f _∗ ([c])f _∗ ² ([c]) · · · f _∗ ^p ⁻ ¹ ([c]) = 1

⇐⇒

[c] = [¯ γ ∗ (f ◦ γ )] in π ₁ (F, x ₀ ) for any path γ from x _i to x ₀ , where x _i ∈ Fix(f )

Remark. If the monodromy f is isotopic to a periodic automorphism g without ﬁxed point, then s _c is hyperbolic for any curve c.

13

(16)

To ﬁnd a hyperbolic section s _c in M _f , say (T _p,q ; 0) explicitly,

we need to recognize which curve c satisﬁes:

Condition

[c]f _∗ ([c]) · · · f _∗ ^p ⁻ ¹ ([c]) = 1 or equivalently

[c] = α ⁻ ¹ f _∗ (α) for any α ∈ π ₁ (F, x ₀ )

We say that an element [c] ∈ π ₁ (F, x ₀ ) is

non-returnable (w.r.t. f ) if it satisﬁes the above condition.

Question. Assume that [c] = 1 ∈ π ₁ (F, x ₀ ).

Then is [c] or [c] ⁻ ¹ non-returnable?

(17)

Partial answer to Question.

Length function of π ₁ (F, x ₀ )

Choose an f -invariant hyperbolic metric on F .

H

²

x₀

~ a~ g_a

x₀

F

a

p

₁

(F, x )

₀

R

[ ]a length(g_a)

L :

Note that L(α ⁻ ¹ ) = L(α).

15

(18)

Theorem 3 Suppose that f : F → F has period p > 2.

Then there is a constant C _p depending on the period p so that

if L([c]) > C _p , then [c] or [c] ⁻ ¹ is non-returnable (w.r.t. f ).

Picking the constant C _p for an irreducible, periodic monodromy f of period p, we have:

Corollary 4 If L([c]) > C _p , then the section

s _c or s _¯ _c is hyperbolic in M _f .

(19)

More precisely, considering the angle from c(0) to ˙ ˙ c(1), we can detect s _c is hyperbolic or s _¯ _c is hyperbolic.

By a numerical computation, we have the fol- lowing table of the constants C _p (3 ≤ p ≤ 15).

p C _p

3 4 5 6 7 8 9 10 11 12 13 2.6 3.2 3.7 4.1 4.4 4.6 4.9 5.1 5.3 5.5 5.6

17

(20)

Example –Hyperbolic section in (T ₂ _, ₅ ; 0)

In the initial construction, assume that (p, q) = (2, 5).

Let us choose a curve c on the ﬁber surface so that L([c]) > 5.1.

Then a section s _c or s _¯ _c is hyperbolic in (T _2,5 ; 0).

s f

x₀ F

c: L([c]) > 5.1 (T_2,5; 0)

hyperbolic section s

f

x₀ F

or -

(21)

Nielsen-Thurston types of the m onodromy f Conditions on c for s b eing hyperbolic

c

Automorphisms of surfaces and Seifert ﬁbered surgeries on ﬁbered knots

Automorphisms of surfaces and Seifert ﬁbered surgeries

on ﬁbered knots

Kimihiko Motegi (Nihon Univ.) joint work with

Kazuhiro Ichihara (Osaka Sangyo Univ.)

Exceptional surgeries on hyperbolic, ﬁbered knots

K : ﬁbered knot in a closed 3-manifold M .

E (K ) = ˆ F × [0, 1]/(x, 0) = ( ˆ f (x), 1);

mapping torus of a once punctured, com- pact, orientable surface ˆ F via a monodromy map ˆ f : ˆ F → F ˆ .

F

^ f

^

Dehn ﬁllings and

capping oﬀ monodromies

An existence of

a Seifert ﬁbered (resp. toroidal) surgery on a hyperbolic, ﬁbered knot K

⇐⇒

An existence of

a pseudo-Anosov monodromy of ˆ F

whose capped oﬀ monodromy is isotopic to a periodic (resp. reducible) automorphism.

[Thurston], [Otal], [Jaco]

F f

f ^

^

(K;0) Dehn filling

x

capping of f

hyperbolic toroidal

Seifert fibered p s eudo-Anosov reducible

periodic F

Longitudinal, toroidal surgery

There is a pseudo-Anosov monodromy of a hyperbolic, ﬁbered knot K in S 3

whose capped oﬀ monodromy is isotopic to a reducible automorphism,

i.e., (K; 0) is a toroidal manifold. [Gabai]

We can ﬁnd inﬁnitely many such phenomena by Osoinach’s construction.

What can we say about longitudinal, Seifert ﬁbered surgery?

There is no known example of a pseudo-Anosov monodromy of a hyperbolic, ﬁbered knot K in S 3

whose capped oﬀ monodromy is isotopic to a periodic automorphism,

i.e., (K; 0) is a Seifert ﬁber space.

In fact, Teragaito conjectures that there are

“no such examples”.

If the monodromy ˆ f has a prong ≥ 2 singular- ity at the boundary, then the invariant mea- sured singular foliation on ˆ F can be naturally extended to that of the capped oﬀ surface F . [suggestion by J.Los]

capping off

prong = 3 prong = 3

F F

^

Thus we have:

Proposition 1 Let K be a hyperbolic, fibered knot in a closed 3-manifold

with a monodromy isotopic to a pseudo-Anosov automorphism having a prong n ≥ 2 singu- larity at the boundary.

Then the resulting manifold obtained by Dehn surgery along the fiber slope is hyperbolic.

In particular, it is not a Seifert fiber space.

Examples in homology 3-spheres

For any integer g ≥ 2,

there is a homology 3-sphere M and a hyperbolic, fibered knot K of genus g in M such that a longitudinal surgery on K yields a Seifert ﬁber space.

In other words,

There is a pseudo-Anosov monodromy of a hyperbolic, ﬁbered knot in homology-S 3 whose capped oﬀ monodromy is isotopic to a periodic automorphism.

Construction

T p,q : (p, q)-torus knot in S 3

Assume q > p ≥ 2 and (p, q) = (2, 3).

Then (T p,q ; 0) = (F × I )/ { (x, 0) = (f (x), 1) } ,

where F is a closed surface of genus ( p − 1)( 2 q − 1) and f is a periodic monodromy of period pq ﬁxing a single point x 0 ∈ F .

M f = (T p,q ; 0) is a Seifert ﬁber space.

Choose a hyperbolic section s of M f passing (x 0 , 0) = (x 0 , 1).

s

Attach a solid torus V to M f − intN (s) to ob- tain a homology 3-sphere Σ and a hyperbolic, ﬁbered knot K ⊂ Σ which is a core of V .

Dually, a surgery on K ⊂ Σ along the ﬁber slope yields a Seifert ﬁber space M f .

For a given integer g ≥ 2,

by starting with (2, 2g + 1)-torus knot in the above,

we have the required examples.

The above construction is based on

an existence of a hyperbolic section in M f , i.e., a section with hyperbolic complement.

It should be noted that if (p, q) = (2, 3), then there is no such a section in M f = (T 2 , 3 ; 0).

The construction leads us:

Question

How can we describe the position of

hyperbolic sections in a surface bundle over the circle?

Visualizing sections on a surface

“3-dimensional view → 2-dimensional view”

f t

x

There is a pseudo-Anosov monodromy of a hyperbolic, ﬁbered knot K in S ³

There is no known example of a pseudo-Anosov monodromy of a hyperbolic, ﬁbered knot K in S ³

There is a pseudo-Anosov monodromy of a hyperbolic, ﬁbered knot in homology-S ³ whose capped oﬀ monodromy is isotopic to a periodic automorphism.

T _p,q : (p, q)-torus knot in S ³

Then (T _p,q ; 0) = (F × I )/ { (x, 0) = (f (x), 1) } ,

where F is a closed surface of genus ⁽ ^p ⁻ ¹⁾⁽ ₂ ^q ⁻ ¹⁾ and f is a periodic monodromy of period pq ﬁxing a single point x ₀ ∈ F .

M _f = (T _p,q ; 0) is a Seifert ﬁber space.

Choose a hyperbolic section s of M _f passing (x ₀ , 0) = (x ₀ , 1).

Attach a solid torus V to M _f − intN (s) to ob- tain a homology 3-sphere Σ and a hyperbolic, ﬁbered knot K ⊂ Σ which is a core of V .

Dually, a surgery on K ⊂ Σ along the ﬁber slope yields a Seifert ﬁber space M _f .

an existence of a hyperbolic section in M _f , i.e., a section with hyperbolic complement.

It should be noted that if (p, q) = (2, 3), then there is no such a section in M _f = (T ₂ _, ₃ ; 0).

f : automorphism of F ﬁxing a speciﬁed point x ₀ ∈ F

t : monotone arc in F × [0, 1] connecting (x ₀ , 0) and (x ₀ , 1)

M _f = F × [0, 1]/(x, 0) = (f (x), 1) : mapping

torus which is a surface bundle over S ¹ Then t deﬁnes a section s ⊂ M _f .

The projection c deﬁnes an element [c] ∈ π ₁ (F, x ₀ ).

[c] = [c ] ∈ π ₁ (F, x ₀ ) ⇒ s _c and s _c

s _c is hyperbolic.

[c]f _∗ ([c])f _∗ ² ([c]) · · · f _∗ ^p ⁻ ¹ ([c]) = 1

[c] = [¯ γ ∗ (f ◦ γ )] in π ₁ (F, x ₀ ) for any path γ from x _i to x ₀ , where x _i ∈ Fix(f )

Remark. If the monodromy f is isotopic to a periodic automorphism g without ﬁxed point, then s _c is hyperbolic for any curve c.

To ﬁnd a hyperbolic section s _c in M _f , say (T _p,q ; 0) explicitly,

[c]f _∗ ([c]) · · · f _∗ ^p ⁻ ¹ ([c]) = 1 or equivalently

[c] = α ⁻ ¹ f _∗ (α) for any α ∈ π ₁ (F, x ₀ )

We say that an element [c] ∈ π ₁ (F, x ₀ ) is

Question. Assume that [c] = 1 ∈ π ₁ (F, x ₀ ).

Then is [c] or [c] ⁻ ¹ non-returnable?

Length function of π ₁ (F, x ₀ )

Note that L(α ⁻ ¹ ) = L(α).

Then there is a constant C _p depending on the period p so that

if L([c]) > C _p , then [c] or [c] ⁻ ¹ is non-returnable (w.r.t. f ).

Picking the constant C _p for an irreducible, periodic monodromy f of period p, we have:

Corollary 4 If L([c]) > C _p , then the section

s _c or s _¯ _c is hyperbolic in M _f .

More precisely, considering the angle from c(0) to ˙ ˙ c(1), we can detect s _c is hyperbolic or s _¯ _c is hyperbolic.

By a numerical computation, we have the fol- lowing table of the constants C _p (3 ≤ p ≤ 15).

p C _p

Example –Hyperbolic section in (T ₂ _, ₅ ; 0)

Then a section s _c or s _¯ _c is hyperbolic in (T _2,5 ; 0).