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.

• Chapters1. Introduction2. Examples3. Proof of Theorems and4. Initial assumptions and reductions5. Culler-Shalen theory6. Bending characters of triangle group amalgams7. The proof of Theorem when $F$ is a semi-fibre8. The proof of Theorem when $F$ is a fibre9. Further assumptions, reductions, and background material10. The proof of Theorem when $F$ is non-separating but not a fibre11. 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 exterior15. 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 $I$-bundle17. 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 small19. The case that $F$ separates but is not a semi-fibre, $t_1^+ = t_1^- = 0$, $d>1$, and $M(\alpha )$ is not very small20. Proof of Theorem

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.

• Chapters1. Introduction2. Examples3. Proof of Theorems and4. Initial assumptions and reductions5. Culler-Shalen theory6. Bending characters of triangle group amalgams7. The proof of Theorem when $F$ is a semi-fibre8. The proof of Theorem when $F$ is a fibre9. Further assumptions, reductions, and background material10. The proof of Theorem when $F$ is non-separating but not a fibre11. 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 exterior15. 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 $I$-bundle17. 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 small19. The case that $F$ separates but is not a semi-fibre, $t_1^+ = t_1^- = 0$, $d>1$, and $M(\alpha )$ is not very small20. Proof of Theorem

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.

• Chapters1. Introduction2. Examples3. Proof of Theorems and4. Initial assumptions and reductions5. Culler-Shalen theory6. Bending characters of triangle group amalgams7. The proof of Theorem when $F$ is a semi-fibre8. The proof of Theorem when $F$ is a fibre9. Further assumptions, reductions, and background material10. The proof of Theorem when $F$ is non-separating but not a fibre11. 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 exterior15. 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 $I$-bundle17. 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 small19. The case that $F$ separates but is not a semi-fibre, $t_1^+ = t_1^- = 0$, $d>1$, and $M(\alpha )$ is not very small20. Proof of Theorem