Networking Seifert surgeries on

Kimihiko Motegi joint work with Arnaud Deruelle and Katura Miyazaki

K: a 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 is a regular fiber in some Seifert fibration of S3.

Wq,p

torusknotTp,q

Vp,q

3 S=Vp,qq,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)pretzelknot (17)=Seifert

(18)=Seifert ()lens

(19)=Seifert ()lens

[Fintushel-Stern]

6 figure-eightknot

(-1)=Seifert

(-2)=Seifert

(-3)=Seifert

[Thurston]

7  Question 1  How can we prove (-1) isaSeifertfiberspace?

 

• 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) isaSeifertfiberspace?

 

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 WhydoIhaveacold?

11 Ihaveacold

12 Ihaveacold

Ihaveacold

13 Ihaveacold

Ihaveacold

Ihaveacold

14 Ihaveacold

Ihaveacold Ihaveacold

Ihaveacold

15 Ihaveacold Ihaveacold

Ihaveacold Ihaveacold

Ihaveacold Ihaveacold

16 Often we can understand the reason why we have cold by considering our “network”.

Ihaveacold Ihaveacold

Ihaveacold Ihaveacold

Ihaveacold Ihaveacold

So it might be possible to understand ....

17 WhyamIaSeifertsurgery?

18 WhyamIaSeifertsurgery?

19 IamaSeifertsurgery

20 IamaSeifertsurgery

IamaSeifertsurgery

21 IamaSeifertsurgery

IamaSeifertsurgery

IamaSeifertsurgery

22 IamaSeifertsurgery

IamaSeifertsurgery IamaSeifertsurgery

IamaSeifertsurgery

23 IamaSeifertsurgery IamaSeifertsurgery

IamaSeifertsurgery IamaSeifertsurgery

IamaSeifertsurgery IamaSeifertsurgery

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 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

Dehnsurgery Dehnsurgery

()-1 2p+2 ()-1 c surgeryalong c thefiber c Seifertfiberspace Seifertfiberspace

31 p’ q r surgery along c c Seifert p q r (lensspace) 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)11 (K,m)22 (K,m)33

Thiscannothappen

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 isconnectedbybasicseiferters

(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 :Seifertsurgeryon torusknot -1

Kn portionofthenetwork

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 Seifertsurgeryonhyperbolicknot hyperbolicseiferters

basicseiferters (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 shortestgeodesic shortestgeodesic (-1)-twist (-1)-twist 1-twist hyperbolicknots hyperbolicknots

shortest shortest shortest basic seiferter

shortest

shortest

Seifertsurgeryonhyperbolic knot Seifertsurgeryontorus knot

46 “Sources” in the Seifert Surgery Network

(K,m)

“source” of(K,m) Seifertsurgeries on torusknots

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 , 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-eightknot

c2

-1 c1 1 2 Seifertsurgerieson Seifertsurgerieson pretzelknotP(-2,3,7) figure-eightknot41

(4,-3)1 (4,-2)1 (4,-1)1

(P,17) (P,18) (P,19) (O,-3) (0,-2) (O,-1)

Seifertsurgerieson sm (T,-8) (T,-7) (T,-6) (T,-5) (T,-3)

trefoilknot -3,2 -3,2 -3,2 -3,2 (T,-4)-3,2 -3,2 (T,-2)-3,2 (T,-1)-3,2 { Seifertsurgeriesontwo-bridgeknots

−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.

torusknot Bergeknot Bergeknot

(K,m) (K,m)

torusknot

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 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 Seifertsurgery Seifertsurgery Seifertsurgery ( ,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)

Seifertsurgeries on torusknots c(K,m)=4

74 Examples

P(-2,3,7) figure-eightknot

c2

-1 c1 1 2 Seifertsurgerieson Seifertsurgerieson pretzelknotP(-2,3,7) figure-eightknot41

(4,-3)1 (4,-2)1 (4,-1)1

(P,17) (P,18) (P,19) (O,-3) (0,-2) (O,-1)

Seifertsurgerieson sm (T,-8) (T,-7) (T,-6) (T,-5) (T,-3) trefoilknot -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