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Dehn Fillings of Knot Manifolds Containing Essential Twice-Punctured Tori

By (author) Steven Boyer, By (author) Cameron McA. Gordon, By (author) Xingru Zhang

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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.

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The Order of Time

Carlo Rovelli
Paperback / softback
224 pages
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Originally published in Italian: L'ordine del tempo (Milan: Adelphi Edizioni, 2017).

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The Coming Wave: AI, Power and Our Future

Mustafa Suleyman
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352 pages
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Helgoland: The Strange and Beautiful Story of Quantum Physics

Carlo Rovelli
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208 pages
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