The maximal tubes under the deformations of a class of 3-dimensional hyperbolic cone-manifolds

Recently, Hodgson and Kerckhoff found a small bound on Dehn surgered 3-manifolds from hyperbolic knots not admitting hyperbolic structures using deformations of hyperbolic cone-manifolds. They asked whether the area normalized meridian length squared of maximal tubular neighborhoods of the singular locus of the cone-manifold is decreasing and that summed with the cone angle squared is increasing as we deform the cone-angles. We confirm this near 0 cone-angles for an infinite family of hyperbolic cone-manifolds obtained by Dehn surgeries along the Whitehead link complements. The basic method is based on explicit holonomy computations using the A-polynomials and finding the maximal tubes. One of the key tool is the Taylor expression of a geometric component of the zero set of the A-polynomial in terms of the cone-angles. We also show a sequence of Taylor expressions for Dehn surgered manifolds converges to one for the limit hyperbolic manifold.Comment: 27 pages, 10 figure

Computing CMB Anisotropy in Compact Hyperbolic Spaces

The measurements of CMB anisotropy have opened up a window for probing the global topology of the universe on length scales comparable to and beyond the Hubble radius. For compact topologies, the two main effects on the CMB are: (1) the breaking of statistical isotropy in characteristic patterns determined by the photon geodesic structure of the manifold and (2) an infrared cutoff in the power spectrum of perturbations imposed by the finite spatial extent. We present a completely general scheme using the regularized method of images for calculating CMB anisotropy in models with nontrivial topology, and apply it to the computationally challenging compact hyperbolic topologies. This new technique eliminates the need for the difficult task of spatial eigenmode decomposition on these spaces. We estimate a Bayesian probability for a selection of models by confronting the theoretical pixel-pixel temperature correlation function with the COBE-DMR data. Our results demonstrate that strong constraints on compactness arise: if the universe is small compared to the `horizon' size, correlations appear in the maps that are irreconcilable with the observations. If the universe is of comparable size, the likelihood function is very dependent upon orientation of the manifold wrt the sky. While most orientations may be strongly ruled out, it sometimes happens that for a specific orientation the predicted correlation patterns are preferred over the conventional infinite models.Comment: 15 pages, LaTeX (IOP style included), 3 color figures (GIF) in separate files. Minor revision to match the version accepted in Class. Quantum Grav.: Proc. of Topology and Cosmology, Cleveland, 1997. The paper can be also downloaded from http://www.cita.utoronto.ca/~pogosyan/cwru_proc.ps.g

Noncyclic covers of knot complements

Hempel has shown that the fundamental groups of knot complements are residually finite. This implies that every nontrivial knot must have a finite-sheeted, noncyclic cover. We give an explicit bound, $\Phi (c)$, such that if $K$ is a nontrivial knot in the three-sphere with a diagram with $c$ crossings and a particularly simple JSJ decomposition then the complement of $K$ has a finite-sheeted, noncyclic cover with at most $\Phi (c)$ sheets.Comment: 29 pages, 8 figures, from Ph.D. thesis at Columbia University; Acknowledgments added; Content correcte

Degenerations of ideal hyperbolic triangulations

Let M be a cusped 3-manifold, and let T be an ideal triangulation of M. The deformation variety D(T), a subset of which parameterises (incomplete) hyperbolic structures obtained on M using T, is defined and compactified by adding certain projective classes of transversely measured singular codimension-one foliations of M. This leads to a combinatorial and geometric variant of well-known constructions by Culler, Morgan and Shalen concerning the character variety of a 3-manifold.Comment: 31 pages, 11 figures; minor changes; to appear in Mathematische Zeitschrif

Prototyping 1,4-butanediol (BDO) biosynthesis pathway in a cell-free transcription-translation (TX-TL) system

Current methods for assembling metabolic pathways require a process of repeated trial and error and have a long design-build-test cycle. Further, it remains a challenge to precisely tune enzyme expression levels for maximizing target metabolite production. Recently it was shown that a cell-free transcriptional-translation system (TX-TL) can be used to rapidly prototype novel complex biocircuits as well as metabolic pathways. TX-TL systems allow protein expression from multiple DNA pieces, opening up the possibility of modulating concentrations of DNA encoding individual pathway enzymes and testing the related effect on metabolite production. In this work, we demonstrate TX-TL as a platform for exploring the design space of metabolic pathways using a 1,4-BDO biosynthesis pathway as an example. Using TX-TL, we verified enzyme expression and enzyme activity and identified the conversion of 4-hydroxybutyrate to downstream metabolites as a limiting step of the 1,4-BDO pathway. We further tested combinations of various enzyme expression levels and found increasing downstream enzyme expression levels improved 1,4-BDO production

In search of Nemesis

The parallax of all stars of visual magnitude greater than about 6.5 has already been measured. If Nemesis is a main-sequence star 1 parsec away, this requires Nemesis's mass to be less than about 0.4 solar masses. If it were less than about 0.05 solar masses its gravity would be too weak to trigger a comet storm. If Nemesis is on the main sequence, this mass range requires it to be a red dwarf. A red dwarf companion would probably have been missed by standard astronomical surveys. Nearby stars are usually found because they are bright or have high proper motion. However, Nemesis's proper motion would now be 0.01 arcsec/yr, and if it is a red dwarf its magnitude is about 10 - too dim to attract attention. Unfortunately, standard four-color photometry does not distinguish between red dwarfs and giants. So although surveys such as the Dearborn Red Star Catalog list stars by magnitude and spectral type, they do not identify the dwarfs. Every star of the correct spectral type and magnitude must be scrutinized. Our candidate list is a hybrid; candidate red stars are identified in the astrometrically poor Dearborn Red Star Catalog and their positions are corrected using the Hubble Guide Star Catalog. When errors in the Dearborn catalog make it impossible to identify the corresponding Hubble star, the fields are split so that we have one centering on each possible candidate. We are currently scrutinizing 3098 fields, which we believe contain all possible red dwarf candidates in the northern hemisphere. Since our last report the analysis and database software has been completely rebuilt to take advantage of updated hardware, to make the data more accessible, and to implement improved methods of data analysis. The software is now completed and we are eliminating stars every clear night