Overview

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.

Full Product Details

Author:   Steven Boyer ,  Cameron McA. Gordon ,  Xingru Zhang
Publisher:   American Mathematical Society
Imprint:   American Mathematical Society
Volume:   Volume: 295 Number: 1469
Weight:   0.272kg
ISBN:  

9781470468705

ISBN 10:   1470468700
Pages:   106
Publication Date:   31 May 2024
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

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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Author Information

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.

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