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9781470468705 Academic Inspection Copy

Dehn Fillings of Knot Manifolds Containing Essential Twice-Punctured Tori

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We show that if a hyperbolic knot manifold M contains an essential twicepunctured torus F with boundary slope ? and admits a filling with slope ? producing a Seifert fibred space, then the distance between the slopes ? and ? is less than or equal to 5 unless M is the exterior of the figure eight knot. The result is sharp; the bound of 5 can be realized on infinitely many hyperbolic knot manifolds. We also determine distance bounds in the case that the fundamental group of the ?-filling contains no non-abelian free group. The proofs are divided into the four cases F is a semi-fibre, F is a fibre, F is non-separating but not a fibre, and F is separating but not a semi-fibre, and we obtain refined bounds in each case.
Steven Boyer, Universite du Quebec a Montreal, Quebec, Canada. Cameron McA. Gordon, University of Texas at Austin, Texas. Xingru Zhang, University at Buffalo, New York.
Chapters 1. Introduction 2. Examples 3. Proof of Theorems and 4. Initial assumptions and reductions 5. Culler-Shalen theory 6. Bending characters of triangle group amalgams 7. The proof of Theorem when $F$ is a semi-fibre 8. The proof of Theorem when $F$ is a fibre 9. Further assumptions, reductions, and background material 10. The proof of Theorem when $F$ is non-separating but not a fibre 11. Algebraic and embedded $n$-gons in $X^\epsilon $ 12. The proof of Theorem when $F$ separates but is not a semi-fibre and $t_1^+ + t_1^- > 0$ 13. Background for the proof of Theorem when $F$ separates and $t_1^+ = t_1^-=0$ 14. Recognizing the figure eight knot exterior 15. Completion of the proof of Theorem when $\Delta (\alpha , \beta ) \geq 7$ 16. Completion of the proof of Theorem when $X^-$ is not a twisted $I$-bundle 17. Completion of the proof of Theorem when ${\Delta }(\alpha ,\beta )=6$ and $d = 1$ 18. The case that $F$ separates but not a semi-fibre, $t_1^+ = t_1^- = 0$, $d \ne 1$, and $M(\alpha )$ is very small 19. The case that $F$ separates but is not a semi-fibre, $t_1^+ = t_1^- = 0$, $d>1$, and $M(\alpha )$ is not very small 20. Proof of Theorem
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