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We construct prime amphicheiral knots that have free period 2. This settles an open question raised by the second-named author, who proved that amphicheiral hyperbolic knots cannot admit free periods and that prime amphicheiral knots cannot admit free periods of order > 2.
We propose and demonstrate a new mask material of AlGaAs native oxide for selective area metalorganic vapor phase epitaxy (MOVPE) which has several advantages over conventional SiNx or SiO2 masks. GaAs selective area growth occurs on masked substrate of AlGaAs native oxide whose Al composition is 0.4, and the wire structures with trapezoidal cross section are formed along ]100] direction on (001) GaAs substrates with line & space mask pattern. Furthermore, after annealing the selectively grown GaAs wire samples, GaAs layers can be regrown with atomically smooth surface, in which GaAs wires are perfectly buried. The results show that this novel selective area MOVPE technique using AlGaAs native oxide masks are promising for quantum nano-structure device fabrication.
This volume is the proceedings of the programme Spaces of Kleinian Groups and Hyperbolic 3-Manifolds held at the Isaac Newton Institute in Cambridge, 21 July–15 August 2003. It is a companion volume to Kleinian Groups and Hyperbolic 3-Manifolds, London Mathematical Society Lecture Notes 299, the proceedings of a conference with the same title held at the Mathematics Institute, University of Warwick, 11–15 September 2001.
The period surrounding these two conferences has seen a series of remarkable advances in our understanding of hyperbolic structures on 3-manifolds. Many of the outstanding issues immediately preceding the Newton Institute meeting related to difficulties in extending results from manifolds with incompressible boundary to the general case. Proofs of Thurston's ending lamination conjecture and the Bers–Sullivan–Thurston density conjecture for general tame groups were announced at the meeting, and the picture was completed not long after the Newton programme, with two independent proofs of Marden's tameness conjecture. As a result, we now have a very clear understanding of the internal geometry of hyperbolic 3-manifolds, combined with an increasingly detailed, but quite intricate, picture of the topology and geometry of the associated deformation spaces of discrete groups.
The Newton Institute meeting turned out to be the international gathering at which many of these new results were disseminated. Almost all the primary contributors took part. Quite how rapid progress has been only became apparent to many of us during the meeting, which will be remembered as a milestone at which all of the new ideas were brought together.
The subject of Kleinian groups and hyperbolic 3-manifolds is currently undergoing explosively fast development, the last few years having seen the resolution of many longstanding conjectures. This volume contains important expositions and original work by some of the main contributors on topics such as topology and geometry of 3-manifolds, curve complexes, classical Ahlfors-Bers theory, computer explorations and projective structures. Researchers in these and related areas will find much of interest here.
Let K be a knot in the 3-sphere S3, N(K) the regular neighbourhood of K and E(K) = cl(S3−N(K)) the exterior of K. The tunnel number t(K) is the minimum number of mutually disjoint arcs properly embedded in E(K) such that the complementary space of a regular neighbourhood of the arcs is a handlebody. We call the family of arcs satisfying this condition an unknotting tunnel system for K. In particular, we call it an unknotting tunnel if the system consists of a single arc.
We give (1) a formula of the first Betti numbers of abelian coverings of links in terms of the Alexander ideals, (2) certain estimates of the orders of the torsion parts of their first homology groups in terms of the Alexander polynomials, and (3) a structure theorem of the first homology groups of -coverings of spatial graphs. As an application, we generalize a result of E. Hironaka on polynomial periodicity of the first Betti numbers in certain towers of abelian coverings of complex surfaces.
The Jones polynomial VL(t) of a link L in S3 contains certain information on the homology of the 2-fold branched covering D(L) of S3 branched along L. The following formulae are proved by Jones and Lickorish and Millett respectively:
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