Networking Seifert surgeries on knots
Kimihiko Motegi joint work with Arnaud Deruelle and Katura Miyazaki Dehn surgery
K: a knot in the 3-sphere S3.
l
N(K) m
remove N(K) g
glue m p g = pm + q l q
p 3 K(q): a 3-manifold obtained from S p by q-Dehn surgery on K
In the following, we consider nontrivial surgery p =6 1 q 0
{ Dehn surgery } = { knots in S3 } × Q
1 Non-hyperbolic surgeries
Let K be a hyperbolic knot in S3, i.e. S3 − K admits a complete hyperbolic metric of finite volume. Q = H ∪ R ∪ T ∪ S ∪ C
H= {r | K(r) is hyperbolic},
R= {r | K(r) is reducible},
T = {r | K(r) is toroidal},
S= {r | K(r) is Seifert } and
C= {r | K(r) is a counterexample to the Ge- ometrization conjecture}.
2 Thurston’s hyperbolic Dehn surgery theory shows: |R ∪ T ∪ S ∪ C| < ∞.
Solution of the Geometrization conjecture implies C= ∅. [Perelman]
The cabling conjecture (Gonz´alez-Acu˜na-Short) asserts R= ∅.
Therefore it is conjectured:
Q = H ∪ T ∪ S for any hyperbolic knot.
3 “Most” knots are hyperbolic, and for “most” hyperbolic knots we have Q = H.
Theorem 1 (Miyazaki-M) An arbitrary knot in S3 can be deformed into a hyperbolic knot with Q = H by a single crossing change. Furthermore, there are infinitely many such crossing changes for the given knot.
4 Surgeries on torus knots
S3 has infinitely many Seifert fibrations and a torus knot is a regular fiber in some Seifert fibration of S3.
Wq,p
torus knot Tp,q
Vp,q
3 S = Vp, q q,pU W
3 The exterior S − intN(Tp,q) is also Seifert fibered. Thus Dehn surgeries on a torus knot naturally create Seifert fiber spaces except for the surgery along the fiber slope.
Moser (1971) conjectured that only surgeries on torus knots can produce Seifert fiber spaces.
5 Seifert surgeries on hyperbolic knots
P(-2,3,7) pretzel knot (17)=Seifert
(18)=Seifert ()lens
(19)=Seifert ()lens
[Fintushel-Stern]
6 figure-eight knot
( -1)=Seifert
(-2)=Seifert
(-3)=Seifert
[Thurston]
7 Question 1 How can we prove ( -1) is a Seifert fiber space?
• Kirby-Rolfsen’s surgery calculus • Montesinos trick • double-primitive, primitive/Seifert-fibered con- structions [Berge], [Dean] • deformation theory of geometric structures may answer the above question.
8 Here is another question:
Question 2 Why ( -1) is a Seifert fiber space?
9 How about the following question in our prac- tical life?
Question 1 How can I show that I have a cold?
Question 2 Why do I have a cold?
10 Why do I have a cold?
11 I have a cold
12 I have a cold
I have a cold
13 I have a cold
I have a cold
I have a cold
14 I have a cold
I have a cold I have a cold
I have a cold
15 I have a cold I have a cold
I have a cold I have a cold
I have a cold I have a cold
16 Often we can understand the reason why we have cold by considering our “network”.
I have a cold I have a cold
I have a cold I have a cold
I have a cold I have a cold
So it might be possible to understand ....
17 Why am I a Seifert surgery?
18 Why am I a Seifert surgery?
19 I am a Seifert surgery
20 I am a Seifert surgery
I am a Seifert surgery
21 I am a Seifert surgery
I am a Seifert surgery
I am a Seifert surgery
22 I am a Seifert surgery
I am a Seifert surgery I am a Seifert surgery
I am a Seifert surgery
23 I am a Seifert surgery I am a Seifert surgery
I am a Seifert surgery I am a Seifert surgery
I am a Seifert surgery I am a Seifert surgery
24 Proposal
Let us look all Seifert surgeries as a network and try to make a global picture of them.
(K (4,1), -127)-11, 13 (K (4, 9), 1)-11, 13
c 1 = (K(-11,13,4), -127) c (P(-2,3,7), 18) T-11, 13 (O, 3)
c2 --1 (T , -7)-3,2 c 4 (K(-11,13,1), -139) (O, 2)
(O, 1)
(T , -22)-11,2 (P(-3,3,5), 1)
(T , -143)-11,13 c’1 (K(-2,0,0), 56) (O, 0) (K(2,0,0), 40)
(T , -6)-3,2 (O, -1)
(C ,(T ), )-2n(n-1)+1, 2 -n, n-1 -4n(n-1)
(T , 15) (O, -2) 3,5 (D(2,7,3), 23) (O, -3) c (T , 6)3,2 (K , 31)2
1 c n (K , 16n-1)n -1
Kn
25 Seifert Surgery Network
A pair (K, m) is a Seifert surgery if K(m) is a Seifert fiber space. (K, m)=(K0, m0) if K and K0 are isotopic in S3 and m = m0. vertices = Seifert surgeries (K, m)’s
-1 -1 -1 ( , ) ( , ) ( , )
figure-eight unknot trefoil edges = ?
26 “seiferters” for Seifert surgeries
s q
K=Tp, q
sm
s p
c (= sµ, sp, sq) enjoys the following property.
• c is a trivial knot in S3.
• c is a Seifert fiber in Tp,q(m) for any integer m =6 pq.
We call sµ, sp and sq basic seiferters.
27 c
c also enjoys the following property.
• c is a trivial knot in S3.
• c is a Seifert fiber in K(−1),K(−2) and K(−3).
We say that c is a seiferter for Seifert surg- eries (K, −1), (K, −2), (K, −3).
28 c
c also enjoys the following property.
• c is a trivial knot in S3.
• c is a Seifert fiber in K(−6),K(−7) and K(−8).
We say that c is a seiferter for Seifert surg- eries (K, −6), (K, −7), (K, −8).
29 seiferter A knot c in S3 − K is called a seiferter for a Seifert surgery (K, m) if c: • is unknotted in S3, • becomes a Seifert fiber in K(m).
K
c
seiferter
Theorem 2 (Miyazaki-M) If a Seifert surgery (K, r) has a seiferter, then r is an integer ex- cept when K is a torus knot or a cable of a torus knot.
“Existence of seiferters” implies the conjecture “Seifert surgery is integral”.
30 “Inheritance” property
3 K =T S 0 -3,2 Kp
2p+2 p-twisting c along c c trefoil
Dehn surgery Dehn surgery
( )-1 2p+2 ( )-1 c surgery along c the fiber c Seifert fiber space Seifert fiber space
31 p’ q r surgery along c c Seifert p q r (lens space) c c 0 q r Seifert lens # lens
“an exceptional fiber of index 0” — degenerate case.
32 edges ⇔ seiferters
Two Seifert surgeries (K, m) and (K0, m0) are connected by an edge if we have a seiferter c such that (K0, m0) is obtained from (K, m) by a single (meaning ±1) twisting along the seiferter c.
K-2 K-1 K -1 -1 -1 ( ) ( , ) ( ) C , C C ,
figure-eight unknot trefoil
33 edges ⇔ seiferters
Two Seifert surgeries (K, m) and (K0, m0) are connected by an edge if we have a seiferter c such that (K0, m0) is obtained from (K, m) by a single (meaning ±1) twisting along the seiferter c.
K-2 K-1 K -1 -1 -1 ( ) ( , ) ( ) C , C C ,
figure-eight unknot-1-twist trefoil
34 edges ⇔ seiferters
Two Seifert surgeries (K, m) and (K0, m0) are connected by an edge if we have a seiferter c such that (K0, m0) is obtained from (K, m) by a single (meaning ±1) twisting along the seiferter c.
K-2 K-1 K -1 -1 -1 ( ) ( , ) ( ) C , C C ,
figure-eight -1-twist unknot-1-twist trefoil
35 edges ⇔ seiferters
Two Seifert surgeries (K, m) and (K0, m0) are connected by an edge if we have a seiferter c such that (K0, m0) is obtained from (K, m) by a single (meaning ±1) twisting along the seiferter c.
K-2 K-1 K -1 -1 -1 ( ) ( , ) ( ) C , C C ,
figure-eight -1-twist unknot-1-twist trefoil
Then we obtain a 1-dimensional complex which we call Seifert Surgery Network.
36 Seiferters, S-families and S-lattices
Each Seifert surgery (K, m) with a seiferter c provides us with infinitely many Seifert surg- eries (Kp, mp) p ∈ Z, which form a subcom- plex consisting of Seifert surgeries– S-family:
Sc(K, m).
K-2 K-1 K -1 -1 -1 ( ) ( , ) ( ) C , C C ,
figure-eight -1-twist unknot-1-twist trefoil
37 Combinatorial structure of the network
Theorem 3 Sc(K, m) forms a 1-dimensional lattice: if (Kp, mp) = (Kq, mq), then p = q.
We call Sc(K, m) an S-lattice.
S-lattice
(K , m )-3 -3 (K , m )-2 -2 (K , m )-1 -1 (K, m) (K , m )1 1 (K , m )2 2 (K , m )3 3
This cannot happen
38 Theorem 4 Any two S-lattices intersect in at most finitely many vertices.
S-lattices
In fact it is expected at most once.
39 Theorem 5 The Seifert Surgery Network is not a tree. In fact, it has infinitely many cycles consisting of Seifert surgeries on hyperbolic knots.
Proposition 6 (O, m) has infinitely many seifer- ters, and hence there are infinitely many edges adjacent to (O, m).
(O, m)
40 Theorem 7 (1) The subnetwork T consist- ing of Seifert surgeries on torus knots and edges corresponding to basic seiferters is con- nected. (2) The subnetwork G consisting of Seifert surgeries on graph knots and edges corre- sponding to basic seiferters for companion torus knots is connected.
(T , 2202)71,31 (T , 2204)71,31
(T , 37)9,4
(T , 280)9,31
T is connected by basic seiferters
(O, 1) (O, 5)=(T , 5)1,4
41 Connectivity problem
Is the Seifert Surgery Network connected? More precisely, for every Seifert surgery (K, m), does there exist a path from (K, m) to a Seifert surgery on a torus knot?
(K (4,1), -127)-11, 13 (K (4, 9), 1)-11, 13
c 1 = (K(-11,13,4), -127) c (P(-2,3,7), 18) T-11, 13 (O, 3)
c2 --1 (T , -7)-3,2 c 4 (K(-11,13,1), -139) (O, 2)
(O, 1)
(T , -22)-11,2 (P(-3,3,5), 1)
(T , -143)-11,13 c’1 (K(-2,0,0), 56) (O, 0) (K(2,0,0), 40)
(T , -6)-3,2 (O, -1)
(C ,(T ), )-2n(n-1)+1, 2 -n, n-1 -4n(n-1)
(T , 15) (O, -2) 3,5 (D(2,7,3), 23) (O, -3) c (T , 6)3,2 (K , 31)2
1 c n (K , 16n-1)n : Seifert surgery on torus knot -1
Kn portion of the network
42 To obtain Seifert surgeries on hyperbolic knots from those on torus knots, we need non-basic seiferters.
Hyperbolic seiferter A seiferter c for (K, m) is said to be hyper- bolic if S3 − K ∪ c is hyperbolic.
(T , 2202)71,31 (T , 2204)71,31 Seifert surgery on hyperbolic knot hyperbolic seiferters
basic seiferters (T , 37)9,4
(T , 280)9,31
(O, 1) (O, 5)=(T , 5)1,4
43 Hyperbolic seiferters and geodesic seiferters
Theorem 8 Let c be a seiferter for (Tp,q, m) which becomes an exceptional fiber. Then, (1) c is a basic seiferter, p+ε (2) q = 2, m = 2p+ε and c is a (1, 2 )-cable of the basic seiferter sp for Tp,2 (ε = ±1), or (3) c is a hyperbolic seiferter for (K, m), i.e. S3 − K ∪ c is hyperbolic.
sp
Tp,q Tp,2
sm sq sp,e
44 Theorem 9 If (K, m) is a Seifert surgery on a hyperbolic knot K with a seiferter c, then it has a hyperbolic seiferter. If c becomes an exceptional fiber, then c is a hyperbolic seiferter.
Corollary 10 If c is a hyperbolic seiferter for
(K, m), then in the S-lattice Sc(K, m), • vertices are Seifert surgeries on hyperbolic knots, and • edges are shortest geodesic seiferters with only finitely many exceptions.
45 figure-eight unknot trefoil
K-2 K-1 K K1
, -1 , -1 C , -1 , -1 ( )C ()C () ()C shortest geodesic shortest geodesic (-1)-twist (-1)-twist 1-twist hyperbolic knots hyperbolic knots
shortest shortest shortest basic seiferter
shortest
shortest
Seifert surgery onhyperbolic knot Seifert surgery ontorus knot
46 “Sources” in the Seifert Surgery Network
(K, m)
“source” of (K, m) Seifert surgeries on torus knots
Which Seifert surgeries on torus knots are “sources”, i.e. can have hyperbolic seiferters?
47 Hyperbolic seiferters for torus knots m-move
a ta m-move
b t’ t a c c c’ t ’c m-slope m-move is symmetric,
m-move
c c’
Two knots c and c0 in S3 − K are said to be m-equivalent if c0 is obtained from c by a finite sequence of m-moves.
48 Proposition 11 Let Tp,q be a nontrivial torus knot and c a seiferter for (Tp,q, m)(m =6 pq) which is an exceptional fiber in some Seifert
fibration of Tp,q(m); if Tp,q(m) is a lens space, we assume that the base surface is S2.
Then c is m-equivalent to a basic seiferter sp, sq or sµ.
This leads us a band connected sum problem.
In fact we can prove:
49 Lemma 12 Let Tp,q be a torus knot with |p| > q ≥ 2.
(1) If a band sum of a basic seiferter sq and 3 Tp,q is a trivial knot in S , then q = 2.
(2) If a band sum of a basic seiferter sp and 3 Tp,q is a trivial knot in S , then (p, q) = (±3, 2).
(3) If a band sum of a basic seiferter sµ and 3 Tp,q is a trivial knot in S , then (p, q) = (±3, 2).
b3
bm s3 cm
s2 b2
Tp, 2 T3, 2 T3, 2
50 Lemma 12 implies:
Proposition 13 For a Seifert surgery (Tp,q, m) (|p| > q > 2), there is no seiferter which is obtained from a basic seiferter by a single m- move.
Candidates for “sources”
Propositions 11 and 13 suggest:
(O, m), (Tp,2, m), (Tp,q, pq ± 1) and (Tp,q, pq) are “sources” in the Seifert Surgery Network.
51 Proposition 14 (1) (O, m) has a hyperbolic seiferter for each integer m. (2) (O, ±1) and (O, 0) have infinitely many hyperbolic seiferters.
(O, m)
Proposition 15 A Seifert surgery (Tp,2, m) (m =6 2p ± 1) has a hyperbolic seiferter.
Proposition 16 A Seifert surgery (Tp,q, pq) (|p| > q ≥ 2) has a hyperbolic seiferter.
52 Neighbors of trefoil
m c1
am s m m+3 m+3 m+3 twist twist twist bm
m+3 m+3 m+2 twist twist twist
m m c = c1
c-6
c-1
m = -1 m = -6
c−1 is a seiferter for (T−3,2, −1) and c−6 is a seiferter for (T−3,2, −6).
53 am-1
s-3 m+2 m+2 m+2 twist twist twist
b-3
m+2 m+2 twist twist
m-1 m c = c2
c-6
c-1
m = -1 m = -6
−1 c is a seiferter for (T−3,2, −2) and −6 c is a seiferter for (T−3,2, −7).
54 m-2 c 3
s 2
m+1 m+1 b2 m+1 twist twist twist
am-2
m+1 m+1 m+2 twist twist twist
m-2 m c = c3
c-6
c-1
m = -1 m = -6
−1 c is a seiferter for (T−3,2, −3) and −6 c is a seiferter for (T−3,2, −8).
55 Portion of the Seifert Surgery Network
(T, -8)
(T, -7)
(T, -6)
(T, -5)
(T, -4)
(T, -3)
(T, -2)
(T, -1)
(T, 0)
sm
56 Portion of the Seifert Surgery Network
(T, -8)
(T, -7)
(T, -6)
(T, -5)
(T, -4)
(T, -3)
(T, -2)
(T, -1)
(T, 0)
sm s p
57 Portion of the Seifert Surgery Network
(T, -8)
(T, -7) s q
(T, -6)
(T, -5)
(T, -4)
(T, -3)
(T, -2)
(T, -1)
(T, 0)
sm s p
58 Portion of the Seifert Surgery Network
(T, -8)
(T, -7) s q
(T, -6)
(T, -5)
(T, -4)
(T, -3)
(T, -2)
(T, -1)
(T, 0)
sm -1 s c p
59 Portion of the Seifert Surgery Network
(T, -8)
(T, -7) s q
(T, -6)
(T, -5)
(T, -4)
c -6 (T, -3)
(T, -2)
(T, -1)
(T, 0)
sm -1 s c p
60 P(-2, 3, 7) figure-eight knot
c2
-1 c1 1 2 Seifert surgeries on Seifert surgeries on pretzel knot P(-2, 3, 7) figure-eight knot 41
(4 , -3)1 (4 , -2)1 (4 , -1)1
(P, 17) (P, 18) (P, 19) (O, -3) (0, -2) (O, -1)
Seifert surgeries on sm (T , -8) (T , -7) (T , -6) (T , -5) (T , -3)
trefoil knot -3, 2 -3, 2 -3, 2 -3, 2 (T , -4)-3, 2 -3, 2 (T , -2)-3, 2 (T , -1)-3, 2 { Seifert surgeries on two-bridge knots
−1 (−2)-twist along c converts T−3,2 into the figure-eight knot and −6 1-twist along c converts T−3,2 into the (−2, 3, 7)-pretzel knot.
61 Lens surgeries
Theorem 17 Let (K, m) be a Berge’s lens surgery on a hyperbolic knot. Then (K, m) is connected to either
• (T2n+ε,n, (2n + ε)n ± 1) for some n, ε = ±1 or
• (Tp,q, pq) for some p, q by a path going along at most two S-lattices.
torus knot Berge knot Berge knot
(K, m) (K, m)
torus knot
62 Application of networking viewpoint
Previously known knots having Seifert surg- eries:
• have strong inversion or cyclic period 2,
• are embedded in genus 2 Heegaard surface of S3.
Question Are these conditions necessary?
63 The next result answers the question in the negative.
Theorem 18 There is an infinite family of hyperbolic knots Kn such that:
(1) Kn admits a Seifert surgery.
(2) Kn has no symmetry.
(3) Kn cannot be embedded in a genus 2 Heegaard surface of S3.
64 “asymmetric” seiferter
Let L be a knot or a link in S3. An orienta- tion preserving diffeomorphism f : S3 → S3 of finite order which leaves L invariant is called a symmetry of L.
A seiferter c for a Seifert surgery (K, m) is said to be symmetric if K ∪ c has a symmetry f : S3 → S3 with f(c) = c, otherwise c is called an asymmetric seiferter.
65 Theorem 19 Let (K, m) be a Seifert surgery on a torus knot or a hyperbolic knot K with an asymmetric seiferter c which becomes an exceptional fiber. Then there is a constant N such that if |n| > N, then
(1) Kn (a knot obtained from K by n-twisting along c) is hyperbolic, (2) c is a unique shortest closed geodesic in 3 S − Kn,
(3) Kn has no symmetry, in particulr, it cannot be embedded in genus 2 Heegaard surface of S3.
66 K K
c1
c’1
• K(1) is a Seifert fiber space
• c1 is a seiferter for (K, 1) which becomes an exceptional fiber in K(1). [Mattman-Miyazaki-M]
0 Lemma 20 (1) c1 is 1-equivalent to the seifer- 0 ter c1, hence c1 is also a seiferter for (K, 1). 0 (2) c1 is asymmetric.
67 0 (1) Apply 1-move to c1 as below.
c1 b c’1
68 Seifert surgery Seifert surgery Seifert surgery ( ,1) ( ,2) ( ,n+1) 1 -1 - n KK 1 Kn
Networking viewpoint enables us to show that
(Kn, n + 1) is a Seifert surgery.
69 Locating Seifert surgeries (Kn, n + 1)
S (P(-3,3,5), 1) S (P(-3,3,5), 1) c 1 c1 c2
c2
P(-3,3,5)
S (P(-3,3,5), 1)c’1
(P(-3, 3, 5), 1)
c’1 S (O, 1) (,)c1 c 2
c 1
o c 2
(O, 1)
70 (K (4,1), -127)-11, 13 (K (4, 9), 1)-11, 13
c 1 = (K(-11,13,4), -127) (P(-2,3,7), 18) (O, 3)
c2 (T , -7)-3,2 (K(-11,13,1), -139) (O, 2)
(O, 1)
(T , -22)-11,2 (P(-3,3,5), 1)
(T , -143)-11,13 c’1 (K(-2,0,0), 56) (O, 0) (K(2,0,0), 40)
(T , -6)-3,2 (O, -1)
(C ,(T ), )-2n(n-1)+1, 2 -n, n-1 -4n(n-1)
(T , 15) (O, -2) 3,5 (D(2,7,3), 23) (O, -3) (T , 6)3,2 (K , 31)2
(K , 16n-1)n
71
(K (4,1), -127)-11, 13 (K (4, 9), 1)-11, 13 = (K(-11,13,4), -127) (P(-2,3,7), 18) (O, 3)
(T , -7)-3,2 (K(-11,13,1), -139) (O, 2)
(O, 1)
(T , -22)-11,2 (P(-3,3,5), 1)
(T , -143)-11,13 (K(-2,0,0), 56) (O, 0) (K(2,0,0), 40)
(T , -6)-3,2 (O, -1)
(C ,(T ), )-2n(n-1)+1, 2 -n, n-1 -4n(n-1)
(T , 15) (O, -2) 3,5 (D(2,7,3), 23) (O, -3) (T , 6)3,2 (K , 31)2
(K , 16n-1)n
72 Distance between Seifert surgeries
distance The distance d((K, m), (K0, m0)) is defined to be the minimal number of S-lattices to connect (K, m) and (K0, m0).
If there is no path from (K, m) to (K0, m0), then d((K, m), (K0, m0)) = ∞.
73 Complexity of Seifert surgeries
Recall that T be the sub-network of Seifert surgeries on torus knots.
complexity The complexity c(K, m) is defined to be the minimum distance to reach T .
(K, m)
Seifert surgeries on torus knots c(K, m) = 4
74 Examples
P(-2, 3, 7) figure-eight knot
c2
-1 c1 1 2 Seifert surgeries on Seifert surgeries on pretzel knot P(-2, 3, 7) figure-eight knot 41
(4 , -3)1 (4 , -2)1 (4 , -1)1
(P, 17) (P, 18) (P, 19) (O, -3) (0, -2) (O, -1)
Seifert surgeries on sm (T , -8) (T , -7) (T , -6) (T , -5) (T , -3) trefoil knot -3, 2 -3, 2 -3, 2 -3, 2 (T , -4)-3, 2 -3, 2 (T , -2)-3, 2 (T , -1)-3, 2
c(K, −1) = c(K, −2) = c(K, −3) = 1 c(P, 17) = c(P, 18) = c(P, 19) = 1 c(Berge’s lens surgery) ≤ 2.
75