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22
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
Geometry of the complex of curves. II. Hierarchical structure
 MW02] [Nag88] [O’N83] [Rie05] [Sar90] [Thu88] [Tro92] Howard Masur and Michael
"... 2. Complexes and subcomplexes of curves 11 3. Projection bounds 19 4. Tight geodesics and hierarchies 25 ..."
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2. Complexes and subcomplexes of curves 11 3. Projection bounds 19 4. Tight geodesics and hierarchies 25
Automorphism groups of free groups, surface groups and free abelian groups. Problems on mapping class groups and related topics
 301¯D316, Proc. Sympos. Pure Math
, 2006
"... The group of 2 × 2 matrices with integer entries and determinant ±1 can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2dimensional torus. Thus this group is the beginning of three natural sequences of ..."
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The group of 2 × 2 matrices with integer entries and determinant ±1 can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2dimensional torus. Thus this group is the beginning of three natural sequences of groups, namely the general linear groups GL(n, Z), the groups Out(Fn) of outer automorphisms of free groups of rank n ≥ 2, and the mapping class groups Mod ± (Sg) of orientable surfaces of genus g ≥ 1. Much of the work on mapping class groups and automorphisms of free groups is motivated by the idea that these sequences of groups are strongly analogous, and should have many properties in common. This program is occasionally derailed by uncooperative facts but has in general proved to be a successful strategy, leading to fundamental discoveries about the structure of these groups. In this article we will highlight a few of the most striking similarities and differences between these series of groups and present some open problems motivated by this philosophy.
Algorithmic detection and description of hyperbolic structures on closed 3manifolds with solvable word problem. Geometry and Topology. Vol 6
, 2002
"... 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.
Arithmetic of hyperbolic 3manifolds
"... This note is an elaboration of the ideas and intuitions of Grothendieck and Weil concerning the “arithmetic topology”. Given 3dimensional manifold M fibering over the circle we introduce an algebraic number field K = Q ( √ d), where d> 0 is an integer number (discriminant) uniquely determined by M ..."
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This note is an elaboration of the ideas and intuitions of Grothendieck and Weil concerning the “arithmetic topology”. Given 3dimensional manifold M fibering over the circle we introduce an algebraic number field K = Q ( √ d), where d> 0 is an integer number (discriminant) uniquely determined by M. The idea is to relate geometry of M to the arithmetic of field K. On this way, we show that V ol M is a limit density of ideals of given norm in the field K (Dirichlet density). The second statement says that the number of cusp points of manifold M is equal to the class number of the field K. It is remarkable that both of the invariants can be explicitly calculated for the concrete values of discriminant d. Our approach is based on the Ktheory of noncommutative C ∗algebras coming from measured foliations and geodesic laminations studied by Thurston et al. We apply the elaborated technique to solve the Poincaré conjecture for given class of manifolds. Key words and phrases: algebraic number fields, C ∗algebra, geometric topology AMS (MOS) Subj. Class.: 11R, 46L, 57M. 1
Ktheory of hyperbolic 3manifolds
, 2009
"... The subject of present note are relationships between certain class of noncommutative C ∗algebras and geometry of 3dimensional manifolds which fiber over the circle. We suggest new classification of such manifolds which is based on the Ktheory of a C ∗algebra coming from measured foliations and ..."
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The subject of present note are relationships between certain class of noncommutative C ∗algebras and geometry of 3dimensional manifolds which fiber over the circle. We suggest new classification of such manifolds which is based on the Ktheory of a C ∗algebra coming from measured foliations and geodesic laminations studied by Thurston et al. In the first part of the paper a bijection between surface bundles with the pseudoAnosov monodromy and stationary AF C ∗algebras is established. In the second part, we apply the elaborated calculus (rotation numbers, associated number fields) to spell out conjectures on the volume and Dehn surgery invariants of the 3dimensional manifolds. In particular, our approach implies that the multiplicity of the GromovThurston map M ↦ → V ol M is equal to the class number of a real quadratic field associated to the manifold M. This generalizes Bianchi’s formula ([1]) for the imaginary quadratic fields known since
PROBLEMS IN FOLIATIONS AND LAMINATIONS OF 3–MANIFOLDS
, 2002
"... 1.1. Notation. I will try to use consistent notation throughout, so for instance, F, G denote foliations, Λ denotes a lamination, λ, µ, ν denote leaves, etc. For a given object X in a manifold M, ˜ X will denote the pullback of X to the universal cover ˜ M. 1.2. Attribution. I have tried to credit ..."
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1.1. Notation. I will try to use consistent notation throughout, so for instance, F, G denote foliations, Λ denotes a lamination, λ, µ, ν denote leaves, etc. For a given object X in a manifold M, ˜ X will denote the pullback of X to the universal cover ˜ M. 1.2. Attribution. I have tried to credit questions to their originators. There are certain
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
An Algorithm to Recognise Small Seifert Fiber Spaces
, 2004
"... The homeomorphism problem is, given two compact nmanifolds, is there an algorithm to decide if the manifolds are homeomorphic or not. The homeomorphism problem has been solved for many important classes of 3manifolds especially those with embedded 2sided incompressible surfaces (cf [12], [15],[1 ..."
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The homeomorphism problem is, given two compact nmanifolds, is there an algorithm to decide if the manifolds are homeomorphic or not. The homeomorphism problem has been solved for many important classes of 3manifolds especially those with embedded 2sided incompressible surfaces (cf [12], [15],[16]), which are called Haken manifolds. It is also wellknown that the homeomorphism problem is easily solvable for two 3manifolds which admit geometries in the sense of Thurston [36], [31]. Hence the recognition problem, to decide if a 3manifold has a geometric structure, is a significant problem. The recognition problem has been solved for all geometric classes, except for the class of small Seifert fibered spaces, which either have finite fundamental group or have fundamental groups which are extensions of �by a triangle group and have finite abelianisation. Our aim in this paper is to give an algorithm to recognise these last classes of 3manifolds, i.e to decide if a given 3manifold is homeomorphic to one in this class. A completely different solution has been announced recently by Tao Li [22]. Also Perelman’s announcement of a solution of the geometrisation conjecture would enable a complete solution of the homeomorphism problem; by identifying which geometric structure a given manifold admits. However it is worth noting that practical algorithms for the homeomorphism and recogntion problems, which can be implemented via software, are very useful for experimentation in 3manifold topology. (See for example [5], [39]).