Results 1  10
of
57
Three dimensional manifolds, Kleinian groups and hyperbolic geometry
 Bull. Amer. Math. Soc
, 1982
"... late 19th century, in which Poincaré had a large role, was the uniformization theory for Riemann surfaces: that every conformai structure on a closed oriented surface is represented by a Riemannian metric of constant curvature. For the typical case of negative Euler characteristic (genus greater tha ..."
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Cited by 311 (3 self)
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late 19th century, in which Poincaré had a large role, was the uniformization theory for Riemann surfaces: that every conformai structure on a closed oriented surface is represented by a Riemannian metric of constant curvature. For the typical case of negative Euler characteristic (genus greater than 1) such a metric gives a hyperbolic structure: any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane, and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions. The exceptional cases, the sphere and the torus, have spherical and Euclidean structures. Threemanifolds are greatly more complicated than surfaces, and I think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 (or more)—except perhaps for the fact that so many 3manifolds are beautiful. The situation has changed, so that I feel fairly confident in proposing the 1.1. CONJECTURE. The interior of every compact ^manifold has a canonical
Scalar curvature and geometrization conjectures for 3manifolds
 in Comparison Geometry (Berkeley 1993–94), MSRI Publications
, 1997
"... Abstract. We first summarize very briefly the topology of 3manifolds and the approach of Thurston towards their geometrization. After discussing some general properties of curvature functionals on the space of metrics, we formulate and discuss three conjectures that imply Thurston’s Geometrization ..."
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Cited by 31 (8 self)
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Abstract. We first summarize very briefly the topology of 3manifolds and the approach of Thurston towards their geometrization. After discussing some general properties of curvature functionals on the space of metrics, we formulate and discuss three conjectures that imply Thurston’s Geometrization Conjecture for closed oriented 3manifolds. The final two sections present evidence for the validity of these conjectures and outline an approach toward their proof.
Group Negative Curvature For 3Manifolds With Genuine Laminations
 6577 (electronic). MR 99e:57023
, 1998
"... this paper is to make that remark more precise. v) Thurston's hyperbolization theorem asserts that atoroidal Haken manifolds have hyperbolic structures. Subsequently Bestvina, Feighn [BF] gave an elementary argument establishing group negative curvature for such manifolds. ..."
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Cited by 27 (1 self)
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this paper is to make that remark more precise. v) Thurston's hyperbolization theorem asserts that atoroidal Haken manifolds have hyperbolic structures. Subsequently Bestvina, Feighn [BF] gave an elementary argument establishing group negative curvature for such manifolds.
Interactive Topological Drawing
, 1998
"... The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in threedimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues ..."
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Cited by 21 (1 self)
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The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in threedimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues. Because the domain of application is mathematics, topological drawing is also concerned with the correct representation and display of these objects on a computer. By correctness we mean that the essential topological features of objects are maintained during interaction. We have chosen to limit the scope of topological drawing to knot theory, a domain that consists essentially of one class of object (embedded circles in threedimensional space) yet is rich enough to contain a wide variety of difficult problems of research interest. In knot theory, two embedded circles (knots) are considered equivalent if one may be smoothly deformed into the other without any cuts or selfintersections. This notion of equivalence may be thought of as the heart of knot theory. We present methods for the computer construction and interactive manipulation of a
Heegaard splittings with boundary and almost normal surfaces
, 2001
"... This paper generalizes the definition of a Heegaard splitting to unify the concepts of thin position for 3manifolds [14], thin position for knots [2], and normal and almost normal surface theory [3], [12]. This gives generalizations of theorems of Scharlemann, Thompson, Rubinstein, and Stocking. In ..."
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Cited by 12 (5 self)
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This paper generalizes the definition of a Heegaard splitting to unify the concepts of thin position for 3manifolds [14], thin position for knots [2], and normal and almost normal surface theory [3], [12]. This gives generalizations of theorems of Scharlemann, Thompson, Rubinstein, and Stocking. In the final section, we use this machinery to produce an algorithm to determine the bridge number of a knot, provided thin position for the knot coincides with bridge position. We also present several algorithmic and finiteness results about Dehn fillings with small Heegaard genus.
Normal Surface Qtheory
, 1998
"... We describe an approach to normal surface theory for triangulated 3manifolds which uses only the quadrilateral disk types (Qdisks) to represent a nontrivial normal surface. Just as with regular normal surface theory, interesting surfaces are among those associated with the vertices of the projecti ..."
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Cited by 11 (0 self)
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We describe an approach to normal surface theory for triangulated 3manifolds which uses only the quadrilateral disk types (Qdisks) to represent a nontrivial normal surface. Just as with regular normal surface theory, interesting surfaces are among those associated with the vertices of the projective solution space of this new Qtheory.
KNOTS WITH UNKNOTTING NUMBER 1 AND ESSENTIAL CONWAY SPHERES
, 2006
"... For a knot K in S 3, let T(K) be the characteristic toric suborbifold of the orbifold (S 3, K) as defined by BonahonSiebenmann. If K has unknotting number one, we show that an unknotting arc for K can always be found which is disjoint from T(K), unless either K is an EMknot (of EudaveMuñoz) or ..."
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Cited by 10 (0 self)
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For a knot K in S 3, let T(K) be the characteristic toric suborbifold of the orbifold (S 3, K) as defined by BonahonSiebenmann. If K has unknotting number one, we show that an unknotting arc for K can always be found which is disjoint from T(K), unless either K is an EMknot (of EudaveMuñoz) or (S 3, K) contains an EMtangle after cutting along T(K). As a consequence, we describe exactly which large algebraic knots (i.e. algebraic in the sense of Conway and containing an essential Conway sphere) have unknotting number one and give a practical procedure for deciding this (as well as determining an unknotting crossing). Among the knots up to 11 crossings in Conway’s table which are obviously large algebraic by virtue of their description in the Conway notation, we determine which have unknotting number one. Combined with the work of OzsvathSzabo, this determines the knots with 10 or fewer crossings that have unknotting number one. We show that an alternating, large algebraic knot with unknotting number one can always be unknotted in an alternating diagram. As part of the above work, we determine the hyperbolic knots in a solid torus which admit a nonintegral, toroidal Dehn surgery. Finally, we show that having unknotting number one is invariant under mutation.
Links with no exceptional surgeries
 COMMENT. MATH. HELV
, 2006
"... 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. A similar statement holds for links. We prove this using two arguments, one geometric and one combina ..."
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Cited by 10 (6 self)
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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. A similar statement holds for links. We prove this using two arguments, one geometric and one combinatorial. The combinatorial argument further implies that every link with at least 2 twist regions and at least 6 crossings per twist region is hyperbolic and gives a lower bound for the genus of a link.