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TANGLES, 2-HANDLE ADDITIONS AND DEHN FILLINGS 171
the link given in [GW2, Figure 7.1 (b)] and ∆(α, β) > 3, then ∆(α, β) = 4
by [GW2, Theorem 1.1] and M2(β) is the double branched cover of one the
two tangles given in Figure 7.4 (d) and (e) of [GW2] (see [GW2, Lemma
7.5]). But again one can verify that the double branched cover of each of
these two tangles is not Seifert fibered. So M = M2. Finally, if M = M3
(which is the exterior in S 3 of the link given in [GW2, Figure 7.1 (c)])
and ∆(α, β) > 3, then ∆(α, β) = 5 by [GW2, Theorem 1.1] and M3(β) is
the double branched cover of the tangle given in Figure 7.5 (d) or (e) of
[GW2] (see [GW2, Lemma 7.5]). But again one can verify that the double
branched cover of each of these two tangles is not Seifert fibered.
(a)
(b)
n crossings
n crossings
n crossings
isotopy
(c)
isotopy
Figure 8. Distance three between a Seifert slope and an
annular and toroidal slope.
Example 17. We give here a family of infinitely many hyperbolic manifolds
Nn with ∂Nn consists of two tori such that one of the tori contains two slopes,
distance three apart, one producing a Seifert fibered manifold and the other
producing a manifold containing an essential torus and an essential annulus.
These examples are constructed based on [GW2, Lemma 7.2]. Here are the
details. The manifold M3 mentioned in the proof of Proposition 16 is the
double branched cover of a twice punctured 3-ball X whose branched set
is a set of proper arcs shown in Figure 8 (a) (which is from [GW2, Figure
7.6 (c)]). M3 is hyperbolic with ∂M3 consists of three tori, which we denote
(d)
n+1 crossings