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On Canonical Triangulations of Once-Punctured Torus Bundles and Two-bridge link complements
We prove the hyperbolization theorem for punctured-torus bun- dles and two-bridge links by decomposing them into ideal tetrahedra which are then given hyperbolic structures.
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Guts of Surfaces and the Colored Jones Polynomial
TLDR
We introduce new invariants for links in 3–manifolds, as well as invariants of hyperbolic knots. Expand
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Links with no exceptional surgeries
We show that if a knot admits a prime, twist-reduced diagram with at least 4 twist regions and at least 6 crossings per twist region, then every non-trivial Dehn filling of that knot is hyperbolike.Expand
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The Jones polynomial and graphs on surfaces
TLDR
In this paper we show that the Jones polynomial of any link can be obtained from the Bollobas-Riordan-Tutte poynomial of a certain oriented ribbon graph associated to a link projection. Expand
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Cusp Areas of Farey Manifolds and Applications to Knot Theory
This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured-torus bundles, 4-punctured sphere bundles, and two-bridge link complements. The input for these estimatesExpand
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Quasifuchsian state surfaces
This paper continues our study, initiated in [arXiv:1108.3370], of essential state surfaces in link complements that satisfy a mild diagrammatic hypothesis (homogeneously adequate). For hyperbolicExpand
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Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic 3-manifolds
This paper proves that every finite volume hyperbolic 3-manifold M contains a ubiquitous collection of closed, immersed, quasi-Fuchsian surfaces. These surfaces are ubiquitous in the sense that theirExpand
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Explicit Dehn filling and Heegaard splittings
We prove an explicit, quantitative criterion that ensures the Heegaard surfaces in Dehn fillings behave "as expected." Given a cusped hyperbolic manifold X, and a Dehn filling whose meridian andExpand
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Finite surgeries on three-tangle pretzel knots
We classify Dehn surgeries on (p, q, r) pretzel knots that result in a manifold of finite funda- mental group. The only hyperbolic pretzel knots that admit non-trivial finite surgeries are ( 2,3,7)Expand
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Dehn filling, volume, and the Jones polynomial
Given a hyperbolic 3–manifold with torus boundary, we bound the change in volume under a Dehn filling where all slopes have length at least 2π. This result is applied to give explicit diagrammaticExpand
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