Pseudo-Anosov braids with small entropy and the magic 3-manifold

Transcription

1 Pseudo-Anosov braids with small entropy and the magic 3-manifold E. Kin M. Takasawa Tokyo Institute of Technology, Dept. of Mathematical and Computing Sciences The 5th East Asian School of Knots and Related Topics

2 Pseudo-Anosov braids and the magic manifold ( ) 2/ 16 1 Introduction Pseudo-Anosov mapping classes We consider a mapping class group M(Σ) for an orientable surface Σ. We are interested in the elements of M(Σ), called pseudo-anosov mapping classes When Σ = D n is an n-punctured disk, a mapping class φ M(D n ) is represented by an n-braid on the disk. It makes sense to speak of pseudo-anosov braids.

3 Pseudo-Anosov braids and the magic manifold ( ) 3/ 16 φ is pseudo-anosov mapping torus T(φ) = Σ [0, 1]/ (x,0) (Φ(x),1) is hyperbolic Remark 1. If b is a braid which represents φ M(D n ), then T(φ) is homeomorphic to the braided link b. b b

4 Pseudo-Anosov braids and the magic manifold ( ) 4/ 16 Two invariants of pseudo-anosov mapping classes 1. volume vol(φ) := volue of the mapping torus T(φ) 2. dilatation λ(φ) (= stretch factor of an unstable foliation) 2 entropy ent(φ) = log(λ(φ)) Problem 1. For Σ, determine the minimal entropy and the minimal volume among pseudo-anosov mappisng classes of M(Σ).

5 Pseudo-Anosov braids and the magic manifold ( ) 5/ 16 Entropy vs Volume for pseudo-anosov maps [Kin, Kojima and Takasawa], arxiv: (1) If entropy is small, then volume can not be large. (2) (Experimental result) If the volume is small, then the entropy is small (Computer experiment for Σ = D 6 by M. Takasawa)

6 Pseudo-Anosov braids and the magic manifold ( ) 6/ 16 The magic 3-manifold M magic := exterior of the 3-chain link A surface bundle over the circle (It admits -ly many fiber surfaces.) having the smallest known volume among hyperbolic 3-cusped manifolds. Many 2-cusped hyperbolic manifolds with small volume can be obtained from M magic by a Dehn surgery. Main result Among n-braids whose braided link are homeomorphic to M magic, we decide which braid reaches the minimal entropy fixing strands. We relate such braids to the braids with the smallest known entropy.

7 Pseudo-Anosov braids and the magic manifold ( ) 7/ 16 2 Basic fact Thurston norm Let M be a hyperbolic 3-mfd (possibly with ). Then Thurston introduced a norm function. X T : H 2 (M, M; R) R. for a connected surface F, let χ (F ) := max{0, χ(f )}. for a primitive integral class a H 2 (M, M; R), X T (a) = min{χ (F ) F is emmebed in M such that [F ] = a}.

8 Pseudo-Anosov braids and the magic manifold ( ) 8/ 16 Theorem 1 (Thurston). The unit ball U = {a H 2 (M, M; R) X T (a) 1} is a convex polyhedron. Example 1 (The unit ball for the magic manifold.). Take embedded surfaces F α, F β, F γ in M magic, and let α = [F α ], β = [F β ], γ = [F γ ]. Then the unit ball is the parallelepiped with vertices ±α, ±β, ±γ, ±(α + β + γ).

9 Pseudo-Anosov braids and the magic manifold ( ) 9/ 16 Relation between Thurston norm and fibers Theorem 2 (Thurston). Let M be a surface bundle over S 1 and let F be a fiber of M. Then ; a top dim. face in U such that (1) [F ] is in the open cone with 0 over. (2) conversely, if a is a primitive integral class in the open cone, then the minimal representative F a of a is a fiber of M. Such is called fiber face.

10 Pseudo-Anosov braids and the magic manifold ( ) 10/ 16 (-1,0,0) β axis (0,1,0) (0,0,1) (0,0,-1) (1,0,0) (1,1,1) Fβ γ axis α axis Fα Fγ (0,1,0) (0,0,-1) 0 (1,0,0) αβ plane (1,1,1) γ axis (0,-1,0) (-1,-1,-1) Every face is a fiber face. Take a fiber face. (It is enough to study this fiber face.)

11 3 Result Pseudo-Anosov braids and the magic manifold ( ) 11/ 16 Let H n = {a int(c (Z)) F a is an n-punctured sphere}. Theorem 3. The homology class which reaches the minimal dilatation among H n except n = 4, 6, 8 is as follows. (1) kα + (k 1)β in case n = 2k + 1. (2) (2k + 1)α + (2k 1)β in case n = 4k + 2. (3a) (4k + 3)α + (4k 1)β in case n = 8k + 4. (3b) (4k + 5)α + (4k + 1)β in case n = 8k + 8.

12 Pseudo-Anosov braids and the magic manifold ( ) 12/ 16 Key propositions Proposition 1. Let a = xα + yβ + zγ int(c (Z)) (x y) such that a is primitive. Then the mini. representative F a has genus 0 iff (x, y, z) is either (1), (2) or (3). (1) z = 0 and gcd(x, y) = 1, (2) (x, y, z) = (n + 1, n, n 1) for n 0 (mod 3) and n 2, or (3) (x, y, z) = (2n + 1, n + 1, n) for n 1. Proposition 2. Let P (t 1, t 2, t 3 ) = t 1 t 2 +t 3 +t 1 t 2 t 1 t 3 t 2 t 3. Then the dilatation of xα + yβ + zγ int(c (Z)) is the largest real root of P (t x, t y, t z ).

13 Monodromy Pseudo-Anosov braids and the magic manifold ( ) 13/ 16 Let T m,p be an m-braid T m,p such that T m,p = (112 m 1) p m 2(m 1) 1 (i := σ i ) By forgetting the first strand of T m,p, one defines an (m 1)-braid from, say T m,p. (Note: ent(t m,p) ent(t m,p ))

14 Pseudo-Anosov braids and the magic manifold ( ) 14/ 16 Theorem 4. Let a = xα + yβ int(c (Z)) such that a is primitive. Then the monodromy φ a : F a F a can be described by the braid T x+y+1,p for some p = p(x, y). In particular we have: ŕ n for H n a monodromy 2k + 1 kα + (k 1)β T 2k,2 4k + 2 (2k + 1)α + (2k 1)β T 4k 1,2k 1 8k + 4 (4k + 3)α + (4k 1)β T 8k+3,2k+1 8k + 8 (4k + 5)α + (4k + 1)β T 8k+7,2k+4 ŕ Remark 2. The braids T (m) := T m,p in Theorem 4 and T (m) := T m,p have small entropy. In particular T (6) B 5 has the smallest entropy [Song-Ham]. The braids T (m) coincides with the braids with the smallest known entropy given by Hironaka-Kin and Venzke

15 Pseudo-Anosov braids and the magic manifold ( ) 15/ 16 Conjecture 1. The braid T (m) or the braid T (m+1) entropy among all braids of B m. realize the minimal

16 Pseudo-Anosov braids and the magic manifold ( ) 16/ 16 How to find a fiber surface k k m full twists (a1) (a2) (b1) (b2) Example 2. (A fiber surface with monodromy T 4,2.)