Longitudinal Seifert Fibered Surgeries on Hyperbolic Knots

(1)

Longitudinal Seifert Fibered Surgeries

on Hyperbolic Knots

Kimihiko Motegi (Nihon Univ.) joint work with

Kazuhiro Ichihara (Osaka Sangyo Univ.) and

Hyun-Jong Song (Pukyong National

University)

(2)

Longitudinal exceptional surgeries on hyperbolic, ﬁbered knots

K : hyperbolic, ﬁbered knot in the 3-sphere S ³ .

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

mapping torus of a once punctured, compact, orientable surface ˆ F with a monodromy map f ˆ : ˆ F → F ˆ isotopic to a pseudo-Anosov

automorphism.

^

f

^

(3)

Dehn ﬁllings and capping oﬀ monodromies

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

mapping torus of the capped oﬀ closed, orientable surface F with the capped oﬀ monodromy map f : F → F .

( K ; 0) is a Seifert ﬁber space (resp. toroidal manifold)

⇐⇒

The capped oﬀ monodromy f is isotopic to a periodic

(resp. reducible) automorphism.

[Thurston], [Otal], [Jaco]

(4)

F

f

x₀

E(K)

^

f

^

(K;0) Dehn filling

x₀ capping off

hyperbolic

toroidal

Seifert fibered pseudo-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 Osoinack’s construction.

(6)

Longitudinal, Seifert ﬁbered surgery on a hyperbolic knot

Fact 1. If K admits a longitudinal, Seifert ﬁbered surgery, then K is a ﬁbered knot [Gabai].

Fact 2. If K is a ( p, q )-torus knot T _p,q or a connected sum of two torus knots T p,q T _p,−q , then a longitudinal surgery on K produces a Seifert ﬁber space.

Proposition 1 Let K be a satellite, fibered knot in S ³ (i.e., a fibered knot whose exterior contains an essential torus).

Then no longitudinal surgery on K yields a

(7)

There have been no known examples of hyperbolic, ﬁbered knots in S ³

with longitudinal, Seifert ﬁbered surgeries.

A question of Teragaito

Does there exist a longitudinal Seifert ﬁbered

surgery on a hyperbolic knot in S ³ ?

(8)

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:

(9)

Proposition 2 Let K be a hyperbolic, fibered knot in S ³ with a monodromy isotopic to a pseudo-Anosov automorphism having

a prong n ≥ 2 singularity at the boundary.

Then ( K ; 0) is hyperbolic.

In particular, it is not a Seifert fiber space.

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Theorem 3 There is an infinite family of hyperbolic, fibered knots in S ³ each of which admits a longitudinal Seifert fibered surgery.

From Proposition 2, we see that the mon-

odromies of ﬁbered knots in Theorem 3 are

isotopic to pseudo-Anosov automorphisms with

prong one singularity at the boundary.

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Knots with longitudinal Seifert surgeries

k

t

₁

t

₂

t

₃

- 1 n+1

-(2n+1)+ 1 2n+2

- 1 n

Let K n be a knot obtained from k by the above surgery description.

K n is a trivial knot for n = 0 , − 1 , − 2.

In what follows, assume that n = 0 , − 1 , − 2.

Lemma 4 (1) K _n is a hyperbolic knot.

(2) ( K n ; 0) is a small Seifert fiber space of

type S ² ( | 2 n + 1 |, | 2 n + 3 |, | (2 n + 1)(2 n + 3) | ).

(12)

Boundary slopes and Seifert ﬁbered surgery slopes

A slope γ on ∂E ( K ) is called a boundary slope if a representative of γ is a boundary compo- nent of an essential surface in the exterior E ( K ).

A knot K in S ³ is said to be small if its ex-

terior contains no closed essential surface.

(13)

Let K be a small hyperbolic knot and γ a boundary slope of K .

Theorem 5 (Culler-Gordon-Luecke-Shalen) ( K ; γ ) cannot have a cyclic fundamental group, in particular, ( K ; γ ) is not a lens space.

Question

Can ( K ; γ ) be a small Seifert fiber space (i.e., a Seifert fiber space over S ² with three exceptional fibers)?

Proposition 6 If ( K ; γ ) is a small Seifert fiber

space, then K is a fibered knot and γ is a fiber

slope (i.e., a longitudinal slope).

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For this remaining possibility, since the knots K n given in Theorem 3 turns out to be small, we have:

Corollary 7 There exists a small hyperbolic

knot in S ³ such that ( K ; γ ) is a small Seifert

fiber space for some boundary slope γ .

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Recall that if a hyperbolic, fibered knot K in S ³ admits a longitudinal Seifert fibered surgery, then the dual knot (i.e., the core of the filled solid torus) is a section in a Seifert fibered, surface bundle with hyperbolic com- plement.

f t

x

₀

F

F [0, 1]

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At the beginning of our study, toward finding a longitudinal Seifert fibered surgery on a hy- perbolic knot, we tried to find a section in a Seifert fibered, surface bundle, say ( T p,q ; 0), so that its exterior is hyperbolic and embed- dable in S ³ .

It is interesting to compare this with Os-

oinach’s examples of longitudinal toroidal surg-

eries from such a viewpoint. He starts with

a longitudinal surgery on a connected sum of

two ﬁgure eight knots 4 ₁ 4 ₁ . His construc-

tion shows that there exist inﬁnitely many

sections in (4 ₁ 4 ₁ ; 0) each of whose comple-

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Question 8 Can we describe the positions of hyperbolic sections in a Seifert fibered, sur- face bundle over the circle?

f t

x

₀

F

F [0, 1]

F : orientable, closed surface of genus ≥ 2.

f : automorphism of F ﬁxing a point x 0 ∈ F t : monotone arc in F × [0 , 1] connecting

( x 0 , 0) and ( x 0 , 1)

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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 of s 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

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Theorem 9 Let F be a closed, orientable surface of genus ≥ 2 and f an irreducible, periodic automorphism of period p with

f ( x 0 ) = x 0 for some point x 0 ∈ F . Let s _c be a section in M _f containing ( x 0 , 0) = ( x 0 , 1) whose projection is c . Then the following three conditions are equivalent.

(1) s c is hyperbolic.

(2) [ c ] f _∗ ([ c ]) · · · f _∗ ^p− ¹ ([ c ]) = 1 ∈ π 1 ( F, x 0 ).

(3) [ c ] = [¯ γ ∗ ( f ◦ γ )] in π 1 ( F, x 0 ) for any path γ from x _i to x 0 , where x _i is a fixed point of f .

Remark. If Fix( f ) = {x 0 } , then (3) is simpli-

ﬁed to the condition “[ c ] = α ⁻ ¹ f ∗ ( α ) for any

α ∈ π 1 ( F, x 0 )”.

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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?

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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 ).

(22)

Theorem 10 Let F be a closed, orientable surface of genus ≥ 2 and f a periodic auto- morphism of period p > 2 such that f ( x 0 ) = x 0 . Then there is a constant C p depending on p so that if L ([ c ]) > C p , then [ c ] or [ c ] ⁻ ¹ is non-returnable.

Theorems 9 and 10 imply:

Corollary 11 Let F, f and C p be as in Theo-

rem 9. Then if L ([ c ]) > C _p , then the section

s c or s ¯ c is hyperbolic in M _f .

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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 approximations of the con- stants 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

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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).

f s

x₀ F

c

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

s f

-

x₀ F

c- -

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An element α ∈ π 1 ( F, x 0 ) is said to be

filling if any representative of α intersects every essential simple closed curve in F .

Theorem 12 Let F be a closed, orientable surface of genus ≥ 2 and f a reducible, peri- odic automorphism of period p with f ( x 0 ) = x 0 for some point x 0 ∈ F . Let s _c be a sec- tion in M _f containing ( x 0 , 0) = ( x 0 , 1) whose projection is c . Then the following two con- ditions are equivalent.

Longitudinal Seifert Fibered Surgeries on Hyperbolic Knots

Longitudinal Seifert Fibered Surgeries

on Hyperbolic Knots

Kimihiko Motegi (Nihon Univ.) joint work with

Kazuhiro Ichihara (Osaka Sangyo Univ.) and

Hyun-Jong Song (Pukyong National

University)

Longitudinal exceptional surgeries on hyperbolic, ﬁbered knots

K : hyperbolic, ﬁbered knot in the 3-sphere S 3 .

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

mapping torus of a once punctured, compact, orientable surface ˆ F with a monodromy map f ˆ : ˆ F → F ˆ isotopic to a pseudo-Anosov

automorphism.

f

Dehn ﬁllings and capping oﬀ monodromies

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

mapping torus of the capped oﬀ closed, orientable surface F with the capped oﬀ monodromy map f : F → F .

( K ; 0) is a Seifert ﬁber space (resp. toroidal manifold)

⇐⇒

The capped oﬀ monodromy f is isotopic to a periodic

(resp. reducible) automorphism.

[Thurston], [Otal], [Jaco]

F

f

f

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 Osoinack’s construction.

Longitudinal, Seifert ﬁbered surgery on a hyperbolic knot

Fact 1. If K admits a longitudinal, Seifert ﬁbered surgery, then K is a ﬁbered knot [Gabai].

Fact 2. If K is a ( p, q )-torus knot T p,q or a connected sum of two torus knots T p,q T p,−q , then a longitudinal surgery on K produces a Seifert ﬁber space.

Proposition 1 Let K be a satellite, fibered knot in S 3 (i.e., a fibered knot whose exterior contains an essential torus).

Then no longitudinal surgery on K yields a

There have been no known examples of hyperbolic, ﬁbered knots in S 3

with longitudinal, Seifert ﬁbered surgeries.

A question of Teragaito

Does there exist a longitudinal Seifert ﬁbered

surgery on a hyperbolic knot in S 3 ?

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 2 Let K be a hyperbolic, fibered knot in S 3 with a monodromy isotopic to a pseudo-Anosov automorphism having

a prong n ≥ 2 singularity at the boundary.

Then ( K ; 0) is hyperbolic.

In particular, it is not a Seifert fiber space.

Theorem 3 There is an infinite family of hyperbolic, fibered knots in S 3 each of which admits a longitudinal Seifert fibered surgery.

From Proposition 2, we see that the mon-

odromies of ﬁbered knots in Theorem 3 are

isotopic to pseudo-Anosov automorphisms with

prong one singularity at the boundary.

Knots with longitudinal Seifert surgeries

k

t

t

t

- 1 n+1

-(2n+1)+ 1 2n+2

- 1 n

Let K n be a knot obtained from k by the above surgery description.

K n is a trivial knot for n = 0 , − 1 , − 2.

In what follows, assume that n = 0 , − 1 , − 2.

Lemma 4 (1) K n is a hyperbolic knot.

(2) ( K n ; 0) is a small Seifert fiber space of

type S 2 ( | 2 n + 1 |, | 2 n + 3 |, | (2 n + 1)(2 n + 3) | ).

Boundary slopes and Seifert ﬁbered surgery slopes

A slope γ on ∂E ( K ) is called a boundary slope if a representative of γ is a boundary compo- nent of an essential surface in the exterior E ( K ).

A knot K in S 3 is said to be small if its ex-

terior contains no closed essential surface.

Let K be a small hyperbolic knot and γ a boundary slope of K .

Theorem 5 (Culler-Gordon-Luecke-Shalen) ( K ; γ ) cannot have a cyclic fundamental group, in particular, ( K ; γ ) is not a lens space.

Question

Can ( K ; γ ) be a small Seifert fiber space (i.e., a Seifert fiber space over S 2 with three exceptional fibers)?

Proposition 6 If ( K ; γ ) is a small Seifert fiber

space, then K is a fibered knot and γ is a fiber

K : hyperbolic, ﬁbered knot in the 3-sphere S ³ .

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

Fact 2. If K is a ( p, q )-torus knot T _p,q or a connected sum of two torus knots T p,q T _p,−q , then a longitudinal surgery on K produces a Seifert ﬁber space.

Proposition 1 Let K be a satellite, fibered knot in S ³ (i.e., a fibered knot whose exterior contains an essential torus).

There have been no known examples of hyperbolic, ﬁbered knots in S ³

surgery on a hyperbolic knot in S ³ ?

Proposition 2 Let K be a hyperbolic, fibered knot in S ³ with a monodromy isotopic to a pseudo-Anosov automorphism having

Theorem 3 There is an infinite family of hyperbolic, fibered knots in S ³ each of which admits a longitudinal Seifert fibered surgery.

Lemma 4 (1) K _n is a hyperbolic knot.

type S ² ( | 2 n + 1 |, | 2 n + 3 |, | (2 n + 1)(2 n + 3) | ).

A knot K in S ³ is said to be small if its ex-

Can ( K ; γ ) be a small Seifert fiber space (i.e., a Seifert fiber space over S ² with three exceptional fibers)?

knot in S ³ such that ( K ; γ ) is a small Seifert

Recall that if a hyperbolic, fibered knot K in S ³ admits a longitudinal Seifert fibered surgery, then the dual knot (i.e., the core of the filled solid torus) is a section in a Seifert fibered, surface bundle with hyperbolic com- plement.

At the beginning of our study, toward finding a longitudinal Seifert fibered surgery on a hy- perbolic knot, we tried to find a section in a Seifert fibered, surface bundle, say ( T p,q ; 0), so that its exterior is hyperbolic and embed- dable in S ³ .

two ﬁgure eight knots 4 ₁ 4 ₁ . His construc-

sections in (4 ₁ 4 ₁ ; 0) each of whose comple-

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 .

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

f ( x 0 ) = x 0 for some point x 0 ∈ F . Let s _c be a section in M _f containing ( x 0 , 0) = ( x 0 , 1) whose projection is c . Then the following three conditions are equivalent.

(2) [ c ] f _∗ ([ c ]) · · · f _∗ ^p− ¹ ([ c ]) = 1 ∈ π 1 ( F, x 0 ).

(3) [ c ] = [¯ γ ∗ ( f ◦ γ )] in π 1 ( F, x 0 ) for any path γ from x _i to x 0 , where x _i is a fixed point of f .

ﬁed to the condition “[ c ] = α ⁻ ¹ f ∗ ( α ) for any

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 )

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

Note that L ( α ⁻ ¹ ) = L ( α ).

Theorem 10 Let F be a closed, orientable surface of genus ≥ 2 and f a periodic auto- morphism of period p > 2 such that f ( x 0 ) = x 0 . Then there is a constant C p depending on p so that if L ([ c ]) > C p , then [ c ] or [ c ] ⁻ ¹ is non-returnable.

rem 9. Then if L ([ c ]) > C _p , then the section