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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 Seifert Fibered Surgeries on Hyperbolic Knotsichihara/Research/talks/Slide/akita... · Longitudinal Seifert Fibered Surgeries on Hyperbolic Knots Kimihiko Motegi (Nihon

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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

S3.

E(K) = F × [0,1]/(x,0) = (f(x),1);

mapping torus of a once punctured, compact,

orientable surface F with a monodromy map

f : F → F isotopic to a pseudo-Anosov

automorphism.

FE(K)

f^

^

1

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]

2

F

f

x0

E(K)

f^

^

(K;0)

Dehn filling

x0

capping off

hyperbolic

toroidal

Seifert fibered

pseudo-Anosov

reducible

periodic

F

3

Longitudinal, toroidal surgery

There is a pseudo-Anosov monodromy of a

hyperbolic, fibered knot K in S3

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.

4

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 Tp,q or a

connected sum of two torus knots Tp,q�Tp,−q,

then a longitudinal surgery on K produces a

Seifert fiber space.

Proposition 1 Let K be a satellite, fibered

knot in S3 (i.e., a fibered knot whose exterior

contains an essential torus).

Then no longitudinal surgery on K yields a

small Seifert fiber space.

5

There have been no known examples of

hyperbolic, fibered knots in S3

with longitudinal, Seifert fibered surgeries.

A question of Teragaito

Does there exist a longitudinal Seifert fibered

surgery on a hyperbolic knot in S3?

6

If the monodromy f has a prong ≥ 2 singular-

ity at the boundary, then the invariant mea-

sured singular foliation on 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:

7

Proposition 2 Let K be a hyperbolic, fibered

knot in S3 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.

8

Theorem 3 There is an infinite family of

hyperbolic, fibered knots in S3 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.

9

Knots with longitudinal Seifert surgeries

k

t1 t2

t3

-1

n+1

-(2n+1)+1

2n+2

- 1n

Let Kn be a knot obtained from k by the

above surgery description.

Kn is a trivial knot for n = 0,−1,−2.

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

Lemma 4 (1) Kn is a hyperbolic knot.

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

type S2(|2n +1|, |2n +3|, |(2n +1)(2n +3)|).

10

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 S3 is said to be small if its ex-

terior contains no closed essential surface.

11

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 S2 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).

12

For this remaining possibility, since the knots

Kn given in Theorem 3 turns out to be small,

we have:

Corollary 7 There exists a small hyperbolic

knot in S3 such that (K; γ) is a small Seifert

fiber space for some boundary slope γ.

13

Recall that if a hyperbolic, fibered knot K

in S3 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.

tf

x0

F

F [0, 1]

14

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 (Tp,q; 0),

so that its exterior is hyperbolic and embed-

dable in S3.

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 41�41. His construc-

tion shows that there exist infinitely many

sections in (41�41; 0) each of whose comple-

ment is hyperbolic and embeddable in S3.

15

Question 8 Can we describe the positions of

hyperbolic sections in a Seifert fibered, sur-

face bundle over the circle?

tf

x0

F

F [0, 1]

F : orientable, closed surface of genus ≥ 2.

f : automorphism of F fixing a point x0 ∈ F

t : monotone arc in F × [0,1] connecting

(x0,0) and (x0,1)

16

s = t /f

x0

Fprojection of s

f

M f

c:

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

torus, which is a surface bundle over S1

Then t defines a section s ⊂ Mf .

The projection c of s defines an element

[c] ∈ π1(F, x0).

[c] = [c′] ∈ π1(F, x0) ⇒ sc and sc′ are isotopic.

Question� �

Can we describe hyperbolic sections by

their “projections” on the surface F?� �

17

Theorem 9 Let F be a closed, orientable

surface of genus ≥ 2 and f an irreducible,

periodic automorphism of period p with

f(x0) = x0 for some point x0 ∈ F . Let sc be

a section in Mf containing (x0,0) = (x0,1)

whose projection is c. Then the following

three conditions are equivalent.

(1) sc is hyperbolic.

(2) [c]f∗([c]) · · · fp−1∗ ([c]) �= 1 ∈ π1(F, x0).

(3) [c] �= [γ ∗ (f ◦ γ)] in π1(F, x0) for any path

γ from xi to x0, where xi is a fixed point of

f .

Remark. If Fix(f) = {x0}, then (3) is simpli-

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

α ∈ π1(F, x0)”.

18

To find a hyperbolic section sc in Mf ,

say (Tp,q; 0), explicitly,

we need to recognize which curve c satisfies:

Condition� �

[c]f∗([c]) · · · fp−1∗ ([c]) �= 1

or equivalently

[c] �= α−1f∗(α) for any α ∈ π1(F, x0)� �

We say that an element [c] ∈ π1(F, x0) is

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

above condition.

Question. Assume that [c] �= 1 ∈ π1(F, x0).

Then is [c] or [c]−1 non-returnable?

19

Partial answer to Question.

Length function of π1(F, x0)

Choose an 〈f〉-invariant hyperbolic metric on

F .

H2

x0~ �

~

g�

x0

F

� (F, x ) R1 0

[ ]� g�length( )

L :

Note that L(α−1) = L(α).

20

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(x0) =

x0. Then there is a constant Cp depending

on p so that if L([c]) > Cp, then [c] or [c]−1 is

non-returnable.

Theorems 9 and 10 imply:

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

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

sc or sc is hyperbolic in Mf .

21

More precisely, considering the angle from

c(0) to c(1), we can detect sc is hyperbolic

or sc is hyperbolic.

By a numerical computation, we have the fol-

lowing table of approximations of the con-

stants Cp (3 ≤ p ≤ 15).

p

Cp

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

22

Example –Hyperbolic section in (T2,5; 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 sc or sc is hyperbolic in (T2,5; 0).

sf

x0

F

c

L([c]) = L([ c ]) > 5.1

(T ; 0)2,5

sf

-

x0

F

c-

-

23

An element α ∈ π1(F, x0) 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(x0) =

x0 for some point x0 ∈ F . Let sc be a sec-

tion in Mf containing (x0,0) = (x0,1) whose

projection is c. Then the following two con-

ditions are equivalent.

(1) sc is hyperbolic.

(2) [c]f∗([c]) · · · fp−1∗ ([c]) ∈ π1(F, x0) is filling.

24

Application to a theory of

surface automorphisms

F

M(F) = { f : F F}isotopy

F = F - int D^

0

f : F F, f(x ) = x , f(D ) = D0 0 0 0

M( F )^

[f]

f’ fisotope

fc

tracing x we obtain a closed curve c0

fc ^[ ]

25

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26