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33
New points of view in knot theory
 Bull. Am. Math. Soc., New Ser
, 1993
"... In this article we shall give an account of certain developments in knot theory which followed upon the discovery of the Jones polynomial [Jo3] in 1984. The focus of our account will be recent glimmerings of understanding of the topological meaning of the new invariants. A second theme will be the c ..."
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In this article we shall give an account of certain developments in knot theory which followed upon the discovery of the Jones polynomial [Jo3] in 1984. The focus of our account will be recent glimmerings of understanding of the topological meaning of the new invariants. A second theme will be the central role that braid
Algorithmic detection and description of hyperbolic structures on closed 3manifolds with solvable word problem
, 2001
"... We outline a rigorous algorithm, first suggested by Casson, for determining whether a closed orientable threemanifold M is hyperbolic, and to compute the hyperbolic structure, if one exists. The algorithm requires that a procedure has been given to solve the word problem in π1M. ..."
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We outline a rigorous algorithm, first suggested by Casson, for determining whether a closed orientable threemanifold M is hyperbolic, and to compute the hyperbolic structure, if one exists. The algorithm requires that a procedure has been given to solve the word problem in π1M.
Ideal triangulations of pseudoAnosov mapping tori. In Topology and geometry in dimension three, volume 560
 of Contemp. Math
, 2011
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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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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.
Linearly bounded conjugator property for mapping class groups. To appear in Geometrical and Functional Analysis, Vol 231
 Department of Mathematics, University of Illinois
, 2013
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A tale of two groups: arithmetic groups and mapping class groups
, 2010
"... In this chapter, we discuss similarities, differences and interaction between two natural and important classes of groups: arithmetic subgroups Γ of Lie groups G and mapping class groups Modg,n of surfaces of genus g with n punctures. We also mention similar properties and problems for related group ..."
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In this chapter, we discuss similarities, differences and interaction between two natural and important classes of groups: arithmetic subgroups Γ of Lie groups G and mapping class groups Modg,n of surfaces of genus g with n punctures. We also mention similar properties and problems for related groups such as outer automorphism groups Out(Fn), Coxeter groups and hyperbolic groups. Since groups are often effectively studied by suitable spaces on which they act, we also discuss related properties of actions of arithmetic groups on symmetric spaces and actions of mapping class groups on Teichmüller spaces, hoping to get across the point that it is the existence of actions on good spaces that makes the groups interesting and special, and it is also the presence of large group actions that also makes the spaces interesting. Interaction between locally symmetric spaces and moduli spaces of Riemann surfaces through the example of the Jacobian map will also be discussed in the last part of this chapter. Since reduction theory, i.e., finding good fundamental domains for proper actions of discrete groups, is crucial to transformation group theory, i.e., to
Computing Triangulations of Mapping Tori of Surface Homeomorphisms
, 2001
"... We present the mathematical background of a software package that computes triangulations of mapping tori of surface homeomorphisms, suitable for Jeff Weeks’s program SnapPea. The package is an extension of the software described in [Bri00]. It consists of two programs. jmt computes triangulations a ..."
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We present the mathematical background of a software package that computes triangulations of mapping tori of surface homeomorphisms, suitable for Jeff Weeks’s program SnapPea. The package is an extension of the software described in [Bri00]. It consists of two programs. jmt computes triangulations and prints them in a humanreadable format. jsnap converts this format into SnapPea’s triangulation file format and may be of independent interest because it allows for quick and easy generation of input for SnapPea. As an application, we obtain a new solution to the restricted conjugacy problem in the mapping class group. 1
SIMPLICIAL STRUCTURES OF KNOT COMPLEMENTS
, 2003
"... It was shown in [5] that there exists an explicit bound for the number of Pachner moves needed to connect any two triangulation of any Haken 3manifold which contains no fibred submanifolds as strongly simple pieces of its JSJdecomposition. In this paper we prove a generalisation of that result to ..."
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It was shown in [5] that there exists an explicit bound for the number of Pachner moves needed to connect any two triangulation of any Haken 3manifold which contains no fibred submanifolds as strongly simple pieces of its JSJdecomposition. In this paper we prove a generalisation of that result to all knot complements. The explicit formula for the bound is in terms of the numbers of tetrahedra in the two triangulations. This gives a conceptually trivial algorithm for recognising any knot complement among all 3manifolds.