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Bridge numbers of links and minimal numbers of meridian generators of link groups

Yeonhee JANG Hiroshima University, JSPS research fellow ———– CONTENTS ———–

• 1. Introduction.
• 2. Result.
• 3. Outline of Proof.
• 4. Open Problems.

• 1. Introduction.
• n-bridge presentation
• n-bridge link : link which admits an n-bridge presentation but

does not admit an (n − 1)-bridge presentation Example) 2-bridge link

• 1. Introduction.

Fact (n-bridge links) (1) The only 1-bridge link is the unknot. (2) (Schubert) completely classified 2-bridge links. (3) (Boileau-Zieschang) determined the bridge numbers of (gener- alized) Montesinos links. (4) (J.) A 3-bridge “arborescent” link is equivalent to one of the following links.

n

β α

• 1

1

β α

• 1

1

′ ′ β α

• 2

2

β α

• 2

2

′ ′ β α

• 1

1

β α

• 1

1

′ ′ β α

• 2

2

β α

• 2

2

′ ′

β α

• 2

2

β α

• 1

1

β α

• 3

3

α0 β α

• 2

2

β α

• 1

1

β α

• 3

3

• b

• 1. Introduction.
• rational tangle : (B3, T)

=

3 + 1 −2 + 1 3 = 12 5

• arborescent tangle : a tangle obtained from rational tangles by the

following operations

T1 T2 T1 T2 T1 T2 T1 T2

• T1 T2
• arborescent link : a link obtained from two arborescent tangles by

gluing their boundaries

• 1. Introduction.
• If a link L ⊂ S3 admits an n-bridge presentation, then its group

π1(S3 \ L) is generated by n meridians. ∵ Note that π1(B1 \ L) = m1, m2, . . . , mn, π1(B2 \ L) = m′

1, m′ 2, . . . , m′ n

and π1(S \ L) = n1, n2, . . . , n2n−1. By van Kampen’s theorem, we have π1(S3 \ L) = m1, m2, . . . , mn, m′

1, m′ 2, . . . , m′ n |

m′

1 = w1(m1, m2, . . . , mn), m′ 1 = w2(m1, m2, . . . , mn), . . .

= m1, m2, . . . , mn | w1(m1, . . . , mn) = w2(m1, . . . , mn), . . . .

• 1. Introduction.

Example

m1 m1

• m2
• m2

2

n2 n1 n3 n4 m1 m1 m

2m1

m m1

2m1

m

2

m1 m1 m

2m1

m

2m1

m

n1 ❀ r1 : m′

1 = m1

n2 ❀ r2 : m′

1 = m1m2m1m2 ¯

m1 ¯ m2 ¯ m1 n3 ❀ r3 : m′

2 = m1m2m1 ¯

m2 ¯ m1 π1(S3 \ L) = m1, m2, m′

1, m′ 2 | r1, r2, r3

= m1, m2 | m1m2m1 = m2m1m2.

• 1. Introduction.
• b(L) : the bridge number of L,
• w(L) : the minimal number of meridians needed to generate the

group π1(S3 \ L) ❀ w(L) ≤ b(L) Question w(L) = b(L)? Fact True when w(L) = 2 (Boileau-Zimmermann), (i.e., w(L) = 2 ⇐ ⇒ b(L) = 2), for generalized Montesinos links (Boileau-Zieschang), for torus links (Rost-Zieschang).

• 2. Result.

Theorem (Boileau-J.) Let L be an arborescent link. Then w(L) = 3 ⇐ ⇒ b(L) = 3.

• 3. Outline of Proof.

Case 1 : M2(L) is a Seifert fibered space ❀ By [Dunbar] and [Burde-Murasugi], L : generalized Montesinos link L : Seifert link L : torus link T L : T its core

3,n 2,n

❀ [Boileau-Zieschang] ❀ [Rost-Zieschang] ❀ 3-bridge!! Case 2 : M2(L) is a graph manifold with a nontrivial JSJ decomposition

✬ ✫ ✩ ✪

Proposition 1 Assume that w(L) = 3 ⇒ τL induces an “inversion” of π1(M2(L)). In particular, if M2(L) is a graph manifold, then g(M2(L)) = 2. ❀ It suffices to show that τL is the “hyper-elliptic” involution asso- ciated with a genus-2 Heegaard splitting of M2(L).

• 3. Outline of Proof.

Case 2 (continued) : M2(L) is a graph manifold with a nontrivial JSJ decomposition

✬ ✫ ✩ ✪

Proposition 2 Let L be an arborescent link such that w(L) = 3. ⇒ L is either hyperbolic or equivalent to the following link.

n

✬ ✫ ✩ ✪

Proposition 3 M : closed ori. 3-mfd with nontrivial JSJ decomp, τ : involution on M s.t. M → M/τ(∼ = S3) : double branched cover / hyperbolic link L. ⇒ (1) τ is hyper-elliptic on each JSJ torus. (2) τ preserves each JSJ piece and each singular fiber.

• 4. Open Problems.

Q1 : w(L) = 3 ⇒ b(L) = 3 ? Q1-1 : Is Q1 true if M2(L) is a graph manifold? (♣ links containing essential tori in their exterior) Q1-2 : Is Q1 true if M2(L) admits a nontrivial JSJ decomp.? (♣ hyperbolic 1-bridge knots in lens spaces) Q2 : w(L) = b(L) ? Q2-1 : When K ⊂ S3 satisfies w(K) = b(K), does Cp,q(K) have the same property? (♣ true when K : 2-bridge knot and (p, q) = (1, 2)) Q2-2 : When Ki ⊂ S3 satisfies w(Ki) = b(Ki) (i = 1, 2), w(K1♯K2) = b(K1♯K2)(= b(K1) + b(K2) − 1)?