1. Introduction ................................................ 1
2. Preliminary lemmas .......................................... 5
3. Γa+ has no interior vertex ................................. 18
4. Possible components of Γa+ ................................. 20
5. The case n1, n2 ≤ 4 ........................................ 26
6. Kleinian graphs ............................................ 34
7. If na = 4, nb ≥ 4 and Γa+ has a small component then Γa is
kleinian ................................................... 37
8. If na = 4, nb ≥ 4 and Γb is non-positive then Γa+ has no
small component ............................................ 41
9. If Γb is non-positive and na = 4 then nb ≤ 4 ............... 46
10. The case n1 = n2 = 4 and Γ1, Γ2 non-positive ................ 51
11. The case na = 4, and Γb positive ........................... 54
12. The case na = 2, nb ≥ 3, and Γb positive ................... 64
13. The case na = 2, nb > 4, Γ1, Γ2, and max(ω1 + ω2, ω3 + ω4)
= 2nb - 2 .................................................. 74
14. The case na = 2, nb > 4, Γ1, Γ2 non-positive, and
ω1 = ω2 = nb ............................................... 78
15. Γa with na ≤ 2 ............................................. 85
16. The case na = 2, nb = 3 or 4, and Γ1, Γ2 non-positive ....... 86
17. Equidistance classes ....................................... 94
18. The case nb = 1 and na = 2 ................................. 96
19. The case n1 = n2 = 2 and Γb positive ....................... 97
20. The case n1 = n2 = 2 and both Γ1, Γ2 non-positive .......... 103
21. The main theorems ......................................... 108
22. The construction of Mi as a double branched cover ......... 111
23. The manifolds Mi are hyperbolic ........................... 122
24. Toroidal surgery on knots in S3 ........................... 131
Bibliography .................................................. 139
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