Asymptotic expansions of Witten–Reshetikhin–Turaev invariants for some simple 3‐manifolds

For any Lie algebra @Fg and integral level k, there is defined an invariant Zk∗(M, L) of embeddings of links L in 3‐manifolds M, known as the Witten–Reshetikhin–Turaev invariant. It is known that for links in S3, Zk∗(S3, L) is a polynomial in q=exp (2πi/(k+c@Fgv), namely, the generalized Jones polynomial of the link L. This paper investigates the invariant Zr−2∗(M,○) when @Fg=@Fs@Fl2 for a simple family of rational homology 3‐spheres, obtained by integer surgery around (2, n)‐type torus knots. In particular, we find a closed formula for a formal power series Z∞(M)∊Q[[h]] in h=q−1 from which Zr−2∗(M,○) may be derived for all sufficiently large primes r. We show that this formal power series may be viewed as the asymptotic expansion, around q=1, of a multivalued holomorphic function of q with 1 contained on the boundary of its domain of definition. For these particular manifolds, most of which are not Z‐homology spheres, this extends work of Ohtsuki and Murakami in which the existence of power series with rational coefficients related to Zk∗(M, ○) was demonstrated for rational homology spheres. The coefficients in the formal power series Z∞(M) are expected to be identical to those obtained from a perturbative expansion of the Witten–Chern–Simons path integral formula for Z∗(M, ○). © 1995 American Institute of Physics.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/69725/2/JMAPAQ-36-11-6106-1.pd

Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial

We study three-dimensional Chern-Simons theory with complex gauge group SL(2,C), which has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds. We show that, in the presence of a single knotted Wilson loop in an infinite-dimensional representation of the gauge group, the classical and quantum properties of such theory are described by an algebraic curve called the A-polynomial of a knot. Using this approach, we find some new and rather surprising relations between the A-polynomial, the colored Jones polynomial, and other invariants of hyperbolic 3-manifolds. These relations generalize the volume conjecture and the Melvin-Morton-Rozansky conjecture, and suggest an intriguing connection between the SL(2,C) partition function and the colored Jones polynomial.Comment: 67 pages, 13 figures, harvma

Torus knots and mirror symmetry

We propose a spectral curve describing torus knots and links in the B-model. In particular, the application of the topological recursion to this curve generates all their colored HOMFLY invariants. The curve is obtained by exploiting the full Sl(2, Z) symmetry of the spectral curve of the resolved conifold, and should be regarded as the mirror of the topological D-brane associated to torus knots in the large N Gopakumar-Vafa duality. Moreover, we derive the curve as the large N limit of the matrix model computing torus knot invariants.Comment: 30 pages + appendix, 3 figure

Order and Stochastic Dynamics in Drosophila Planar Cell Polarity

Cells in the wing blade of Drosophila melanogaster exhibit an in-plane polarization causing distal orientation of hairs. Establishment of the Planar Cell Polarity (PCP) involves intercellular interactions as well as a global orienting signal. Many of the genetic and molecular components underlying this process have been experimentally identified and a recently advanced system-level model has suggested that the observed mutant phenotypes can be understood in terms of intercellular interactions involving asymmetric localization of membrane bound proteins. Among key open questions in understanding the emergence of ordered polarization is the effect of stochasticity and the role of the global orienting signal. These issues relate closely to our understanding of ferromagnetism in physical systems. Here we pursue this analogy to understand the emergence of PCP order. To this end we develop a semi-phenomenological representation of the underlying molecular processes and define a “phase diagram” of the model which provides a global view of the dependence of the phenotype on parameters. We show that the dynamics of PCP has two regimes: rapid growth in the amplitude of local polarization followed by a slower process of alignment which progresses from small to large scales. We discuss the response of the tissue to various types of orienting signals and show that global PCP order can be achieved with a weak orienting signal provided that it acts during the early phase of the process. Finally we define and discuss some of the experimental predictions of the model

Urban Legends and Paranormal Beliefs: The Role of Reality Testing and Schizotypy

Recent research suggests that unconventional beliefs are locatable within a generic anomalous belief category. This notion derives from the observation that apparently dissimilar beliefs share fundamental, core characteristics (i.e., contradiction of orthodox scientific understanding of the universe and defiance of conventional understanding of reality). The present paper assessed the supposition that anomalous beliefs were conceptually similar and explicable via common psychological processes by comparing relationships between discrete beliefs [endorsement of urban legends (ULs) and belief in the paranormal] and cognitive-perceptual personality measures [proneness to reality testing (RT) and schizotypy]. A sample of 222 volunteers, recruited via convenience sampling, took part in the study. Participants completed a series of self-report measures (Urban Legends Questionnaire, Reality Testing subscale of the Inventory of Personality Organization, Revised Paranormal Belief Scale and the Schizotypal Personality Questionnaire Brief). Preliminary analysis revealed positive correlations between measures. Within schizotypy, the cognitive-perceptual factor was most strongly associated with anomalistic beliefs; disorganized and interpersonal produced only weak and negligible correlations respectively. Further investigation indicated complex relationships between RT, the cognitive-perceptual factor of schizotypy and anomalistic beliefs. Specifically, proneness to RT deficits explained a greater amount of variance in ULs, whilst schizotypy accounted for more variance in belief in the paranormal. Consideration of partial correlations supported these conclusions. The relationship between RT and ULs remained significant after controlling for the cognitive-perceptual factor. Contrastingly, the association between the cognitive-perceptual factor and ULs controlling for RT was non-significant. In the case of belief in the paranormal, controlling for proneness to RT reduced correlation size, but relationships remained significant. This study demonstrated that anomalistic beliefs vary in nature and composition. Findings indicated that generalized views of anomalistic beliefs provide only limited insight into the complex nature of belief

The Problem of Experience in the Study of Organizations

This paper deals with the fact that we cannot experience large organizations directly, in the same way as we can experience individuals or small groups, and that this non-experientiability has certain implications for our scientific theories of organizations. Whereas a science is animated by a constructive interplay of theory concepts and experience concepts, the study of organizations has been confined to theory concepts alone. Implications of this analysis for developing a science of organizations are considered.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/68303/2/10.1177_017084069301400102.pd

The PDZ Protein Canoe/AF-6 Links Ras-MAPK, Notch and Wingless/Wnt Signaling Pathways by Directly Interacting with Ras, Notch and Dishevelled

Over the past few years, it has become increasingly apparent that signal transduction pathways are not merely linear cascades; they are organized into complex signaling networks that require high levels of regulation to generate precise and unique cell responses. However, the underlying regulatory mechanisms by which signaling pathways cross-communicate remain poorly understood. Here we show that the Ras-binding protein Canoe (Cno)/AF-6, a PDZ protein normally associated with cellular junctions, is a key modulator of Wingless (Wg)/Wnt, Ras-Mitogen Activated Protein Kinase (MAPK) and Notch (N) signaling pathways cross-communication. Our data show a repressive effect of Cno/AF-6 on these three signaling pathways through physical interactions with Ras, N and the cytoplasmic protein Dishevelled (Dsh), a key Wg effector. We propose a model in which Cno, through those interactions, actively coordinates, at the membrane level, Ras-MAPK, N and Wg signaling pathways during progenitor specification

Wnt, Hedgehog and Junctional Armadillo/β-Catenin Establish Planar Polarity in the Drosophila Embryo

To generate specialized structures, cells must obtain positional and directional information. In multi-cellular organisms, cells use the non-canonical Wnt or planar cell polarity (PCP) signaling pathway to establish directionality within a cell. In vertebrates, several Wnt molecules have been proposed as permissible polarity signals, but none has been shown to provide a directional cue. While PCP signaling components are conserved from human to fly, no PCP ligands have been reported in Drosophila. Here we report that in the epidermis of the Drosophila embryo two signaling molecules, Hedgehog (Hh) and Wingless (Wg or Wnt1), provide directional cues that induce the proper orientation of Actin-rich structures in the larval cuticle. We further find that proper polarity in the late embryo also involves the asymmetric distribution and phosphorylation of Armadillo (Arm or β-catenin) at the membrane and that interference with this Arm phosphorylation leads to polarity defects. Our results suggest new roles for Hh and Wg as instructive polarizing cues that help establish directionality within a cell sheet, and a new polarity-signaling role for the membrane fraction of the oncoprotein Arm

Experimental and theoretical evidence for bidirectional signaling via core planar polarity protein complexes in Drosophila

In developing tissues, sheets of cells become planar polarized, enabling coordination of cell behaviors. It has been suggested that “signaling” of polarity information between cells may occur either bidirectionally or monodirectionally between the molecules Frizzled (Fz) and Van Gogh (Vang). Using computational modeling we find that both bidirectional and monodirectional signaling models reproduce known non-autonomous phenotypes derived from patches of mutant tissue of key molecules but predict different phenotypes from double mutant tissue, which have previously given conflicting experimental results. Furthermore, we re-examine experimental phenotypes in the Drosophila wing, concluding that signaling is most likely bidirectional. Our modeling suggests that bidirectional signaling can be mediated either indirectly via bidirectional feedbacks between asymmetric intercellular protein complexes or directly via different affinities for protein binding in intercellular complexes, suggesting future avenues for investigation. Our findings offer insight into mechanisms of juxtacrine cell signaling and how tissue-scale properties emerge from individual cell behaviors