Christopher M. Herald

Gauge theoretic invariants of Dehn surgeries on knots

New methods for computing a variety of gauge theoretic invariants for homology 3{spheres are developed. These invariants include the Chern{Simons invariants, the spectral flow of the odd signature operator, and the rho invariants of irreducible SU(2) representations. These quantities are calculated for flat SU(2) connections on homology 3{spheres obtained by 1=k Dehn surgery on (2;q) torus knots. The methods are then applied to compute the SU(3) gauge theoretic Casson invariant (introduced in [5]) for Dehn surgeries on (2;q ) torus knots for q =3 ; 5 ; 7a nd 9.

The pillowcase and perturbations of traceless representations of knot groups

We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowkas singular instanton knot homology non-degenerate. The mechanism for this study is a (Lagrangian) intersection diagram which arises, through restriction of representations, from a tangle decomposition of a knot. When one of the tangles is trivial, our perturbations allow us to study isolated intersections of two Lagrangians to produce minimal generating sets for singular instanton knot homology. The (symplectic) manifold where this intersection occurs corresponds to the traceless character variety of the four-punctured 2-sphere, which we identify with the familiar pillowcase. We investigate the image in this pillowcase of the traceless representations of tangles obtained by removing a trivial tangle from 2-bridge knots and torus knots. Using this, we compute the singular instanton homology of a variety of torus knots. In many cases, our computations allow us to understand non-trivial differentials in the spectral sequence from Khovanov homology to singular instanton homology.

Simulation of Aggregates with Point-Contacting Monomers in the Cluster–Dilute Regime. Part 1: Determining the Most Reliable Technique for Obtaining Three-Dimensional Fractal Dimension from Two-Dimensional Images

Analysis of electron microscopy images of fractal-like aggregates involves extraction of three-dimensional (3-d) structural and geometrical properties of aggregates, which are commonly unknown, from their two-dimensional (2-d) projected images. The fractal dimension Df of an aggregate is considered to be the key property for characterizing fractal-like aggregates. The nested squares method (NSM) (also known as the cumulative-intersection method and concentric circles method), the perimeter grid method (PGM), and the ensemble method (EM) have found wide use as techniques for determination of Df of both individual and ensemble aggregates in the cluster-dilute regime. However, no study has so far compared the validity and accuracy of these three most commonly used analysis methods. In this article, using the fractal simulation package FracMAP, these methods were individually tested by applying them to a statistically significant (∼2500 per fractal dimension) number of projected images of all stable orientations of computer-generated 3-d fractal aggregates with Df ranging between 1.0 and 3.0 in increments of 0.1. Results show that of the three methods, the only method that can be used to reliably determine Df from 2-d images is the EM. Both the NSM and the PGM yield many overlapping values of 2-d Df for differing values of 3-d Df resulting in a non-one-to-one relationship and large margins of error. A correction factor has been formulated as a piece-wise function of linear functions for calibrating EM measured values of 2-d Df to actual 3-d Df values.

Simulation of Aggregates with Point-Contacting Monomers in the Cluster-Dilute Regime. Part 2: Comparison of Two- and Three-Dimensional Structural Properties as a Function of Fractal Dimension

In the past two decades, several experimental and simulation studies have proposed simple empirical relations between projected two-dimensional (2-D) and three-dimensional (3-D) structural properties of fractal-like aggregates in the cluster–dilute regime. These empirical relations have found extensive use in inferring the 3-D structural properties of aggregates from their projected (i.e., 2-D) properties—measurable from aggregate electron microscopy images. This study probes the limitations and nongeneralizability of these simple and straightforward empirical relationships and proposes replacing them with new empirical formulas. A straightforward empirical relationship for directly determining the 3-D fractal dimension (Df ) of an aggregate from the knowledge of its 2-D aspect ratio is also identified. These new relationships were derived by comparing the ratios of several 2-D and 3-D structural properties of a statistically significant number of simulated aggregates with point-contacting monomers as a function of their 3-D Df ranging from 1.0 to 3.0 in increments of 0.1.

FracMAP: A user-interactive package for performing simulation and orientation-specific morphology analysis of fractal-like solid nano-agglomerates !

Abstract Computer simulation techniques have found extensive use in establishing empirical relationships between three-dimensional (3d) and two-dimensional (2d) projected properties of particles produced by the process of growth through the agglomeration of smaller particles (monomers). In this paper, we describe a package, FracMAP, that has been written to simulate 3d quasi-fractal agglomerates and create their 2d pixelated projection images by restricting them to stable orientations as commonly encountered for quasi-fractal agglomerates collected on filter media for electron microscopy. Resulting 2d images are analyzed for their projected morphological properties. Program summary Program title: FracMAP Catalogue identifier: AEDD_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEDD_v1_0.html Program obtainable from: CPC Program Library, Queens University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 4722 No. of bytes in distributed program, including test data, etc.: 27 229 Distribution format: tar.gz Programming language: C++ Computer: PC Operating system: Windows, Linux RAM: 2.0 Megabytes Classification: 7.7 Nature of problem: Solving for a suitable fractal agglomerate construction under constraints of typical morphological parameters. Solution method: Monte Carlo approximation. Restrictions: Problem complexity is not representative of run-time, since Monte Carlo iterations are of a constant complexity. Additional comments: The distribution file contains two versions of the FracMAP code, one for Windows and one for Linux. Running time: 1 hour for a fractal agglomerate of size 25 on a single processor.

Porosity, Specific Surface Area and Effective Thermal Conductivity of Anisotropic Open Cell Lattice Structures

Open-cell box-lattice structures consisting of mutually orthogonal thermally conductive cylindrical ligaments can be configured to have wide ranging porosity, a large specific surface area and effective thermal conductivity in a particular direction together with specified structural characteristics. Thermal and mechanical properties can be tuned (and anisotropy introduced) by specification of different filament diameter and pitch for the vertical and horizontal filaments. Analytical models for porosity, specific surface area and effective thermal conductivity of lattice structures having different ligament diameters and pitches (anisotropy) are developed. The models show that all three of these quantities are functions of three dimensionless lengths. This paper was also originally published as part of the Proceedings of the ASME 2005 Heat Transfer Summer Conference.Copyright

Thermal/Fluid Characteristics of Elliptic Cross Section Filament Box Lattice Matrices as Heat Exchanger Surfaces

‡The thermal and fluid characteristics of elliptic cross section filament box lattices are studied. Analytical models for porosity and specific surface area are developed and are expressed in terms of six dimensionless parameters. Numerical simulations are performed to evaluate the flow and thermal characteristics of these structures deployed as heat exchanger matrices. The flows considered are laminar with Reynolds number ranging from 30 to 150. Friction factors and Stanton numbers are calculated from the simulated flow fields. The effect of streamlining the filaments via ellipticity is discussed by comparison with circular cross section filament box-lattices and three-dimensional filament woven structures.

The SU(2) Casson–Lin invariant of the Hopf link

We compute the

THE SU(3) CASSON INVARIANT FOR INTEGRAL HOMOLOGY 3-SPHERES

SU(2)

An integer valued SU(3) Casson invariant

Casson-Lin invariant for the Hopf link and determine the sign in the formula of Harper and Saveliev relating this invariant to the linking number.