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91
Three dimensional manifolds, Kleinian groups and hyperbolic geometry
 BULL. AMER. MATH. SOC
, 1982
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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 40 (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.
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 35 (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.
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 31 (3 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
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 23 (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.
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 19 (10 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.
Hardness of embedding simplicial complexes in R^d
, 2009
"... Let EMBEDk→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into d? Known results easily imply polynomiality of EMBEDk→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDk→2k for a ..."
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Cited by 15 (5 self)
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Let EMBEDk→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into d? Known results easily imply polynomiality of EMBEDk→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDk→2k for all k ≥ 3. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5sphere implies that EMBEDd→d and EMBED (d−1)→d are undecidable for each d ≥ 5. Our main result is NPhardness of EMBED2→4 and, more generally, of EMBEDk→d for all k, d with d ≥ 4 and d ≥ k ≥ (2d −2)/3. These dimensions fall outside the metastable range of a theorem of Haefliger and Weber, which characterizes embeddability using the deleted product obstruction. Our reductions are based on examples, due to Segal, Spie˙z, Freedman, Krushkal, Teichner, and Skopenkov, showing that outside the metastable range the deleted product obstruction is not sufficient to characterize embeddability.
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 15 (9 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.