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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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