Let M be a cpt ori 3-mfd with ∂M ' T 2 .

Distances between boundary slopes of immersed essential surfaces

市原一裕

Kazuhiro Ichihara

大阪産業大学教養部

Osaka Sangyo Univ.

§ 1. Introduction

Let M be a cpt ori 3-mfd with ∂M ' T ² .

Def. (immersed essential surface)

F is an immersed surface in M

⇐⇒ ∃ def compact surf S with ∂S 6 = ∅

∃ proper immersion f : S ^# M

s.t. F = f (S ), f | _∂S is an embedding.

F is essential

⇐⇒ def f is π ₁ -injective & ∂ - π ₁ -injective.

Plenty of immersed ess surf’s

slope ⇐⇒ ^def isotopy class of s.c.c. on ∂M .

∂ -slope of immersed surf F

⇐⇒ def slope determined by ∂F .

Fact (Maher)

For a two-bridge knot exterior,

all slopes are ∂ -slopes of immersed ess surf.

Question

Can we have any estimation of

∂ -slopes of immersed ess surf ?

§ 2. Distances between ∂ -slopes

Def (Distance between slopes)

Let r ₁ , r ₂ be two slopes on ∂M .

Distance ∆( r ₁ , r ₂ ) ⇐⇒ ^def minimal geometric intersection number of their representative.

Notation:

χ ( F ) := Euler characteristic of S .

∂F := f (∂S ).

# ∂F := number of conn compo of ∂F .

Fact (Hass-Rubinstein-Wang)

Let M be a cpt ori 3-mfd with ∂M ' T ² s.t. intM is complete hyperbolic.

Let F _i be immersed essential surf in M with ∂ -slope r _i for i = 1 , 2.

⇒ ∆(r ₁ , r ₂ ) < ⁴³ ₄ · ^−χ(F _{# ∂F} ^{1 )}

1 · ^−χ(F _{# ∂F} ^{2 )}

2

In particular, if F _i ’s are orientable,

∆( r ₁ , r ₂ ) < ⁴³ ₄ · genus( F ₁ ) · genus( F ₂ )

§ 3. Results

Let F ₁ , F ₂ be immersed essential surf in M with ∂ -slopes r ₁ , r ₂ .

Theorem 1.

If M is Seifert fibered, then

∆( r ₁ , r ₂ ) ≤ 2









−χ(F ₁ )

# ∂F ₁ + −χ(F ₂ )

# ∂F ₂







 + 4 In particular, if F _i ’s are orientable,

∆( r ₁ , r ₂ ) ≤ 4(genus( F ₁ ) + genus( F ₂ ))

Theorem 2.

If intM is hyperbolic and r _i ’s are integral w.r.t. some meri-longi system on ∂M ,

∆( r ₁ , r ₂ ) < 6









−χ ( F ₁ )

# ∂F ₁ + −χ ( F ₂ )

# ∂F ₂









In particular, if F _i ’s are orientable,

∆( r ₁ , r ₂ ) < 12(genus( F ₁ ) + genus( F ₂ ) − 1)

Compare to (H-R-W)’s

∆(r ₁ , r ₂ ) < ⁴³ ₄ · genus(F ₁ ) · genus(F ₂ )

Theorem 3.

If M is a knot exterior in S ³ and F _i ’s are immersed spanning surface without triple points, then

∆( r ₁ , r ₂ ) ≤ 2









−χ ( F ₁ )

# ∂F ₁ + −χ ( F ₂ )

# ∂F ₂







 + 4

Example

K = (−2 , 3 , n )-pretzel knot

(n = 7, 9, 11, . . .)

∃F _i : embedded essential surf

with ∂ -slope r _i for i = 1 , 2 s.t.

−χ(F ₁ ) #∂F ₁ r ₁ −χ(F ₂ ) #∂F ₂ r ₂

n − 6 1 16 n − 5 2 ₍ ⁿ _n− ^{2 −n−5} _{3) / 2} Then ∆( r ₁ , r ₂ ) = n ² + 7 n − 29;

quadratic with respect to genera

Theorem 4.

If M is a small Seifert fibered space, then









−χ ( F ₁ )

#∂F ₁ + −χ ( F ₂ )

#∂F ₂







 + 2 ≤ ∆( r ₁ , r ₂ )

New Result (Advertisement)

Theorem.

Let M be a cpt ori 3-mfd with ∂M '

T ² s.t. int M is complete hyperbolic. If

two slopes r ₁ , r ₂ are both integral slopes

w.r.t. some meri-longi system on ∂M , and

both r ₁ -, r ₂ -surgeries yield non-hyperbolike

manifolds, then ∆(r ₁ , r ₂ ) ≤ 8. Thus

there are at most NINE such integral non-

hyperbolike surgeries.