Octahedral developing of knot complement II: Ptolemy coordinates and applications

It is known that a knot complement (minus two points) decomposes into ideal octahedra with respect to a given knot diagram. In this paper, we study the Ptolemy variety for such an octahedral decomposition in perspective of Thurston's gluing equation variety. More precisely, we compute explicit Ptolemy coordinates in terms of segment and region variables, the coordinates of the gluing equation variety motivated from the volume conjecture. As a consequence, we present an explicit formula for computing the obstruction to lifting a $(\mathrm{PSL}(2,\mathbb{C}),P)$-representation of the knot group to a $(\mathrm{SL}(2,\mathbb{C}),P)$-representation. We also present a diagrammatic algorithm to compute a holonomy representation of the knot group.Comment: 32 pages, 21 figue

Adjoint Reidemeister torsions from wrapped M5-branes

We introduce a vanishing property of adjoint Reidemeister torsions of a cusped hyperbolic 3-manifold derived from the physics of wrapped M5-branes on the manifold. To support our physical observation, we present a rigorous proof for the figure-eight knot complement with respect to all slopes. We also present numerical verification for several knots

Simulation of Magnetic Fluid to Develop the Magnetic Chromatography for Magnetic Particle Separation

A magnetic chromatography is a very useful system for an ion and/or fine magnetic particle separation. Recently, its performance has been enhanced using a superconducting magnet. The superconducting magnet can generate a high magnetic field and its strong magnetic field gradients in a very small flow channel. We have developed the magnetic chromatography system to separate the fine particles. However, its numerical simulation is difficult, since the scale of the fine magnetic particles in fluid is much different from the scale of the superconducting magnet generating the strong magnetic field. In order to accurately simulate the magnetic separation, it is necessary to develop the simulation code dealing with the multi-scale. The performance of the developed magnetic chromatography is evaluated by the developed simulation tool and it is clarified that it is unsuitable for magnetic particle separation. Therefore, a magnetic chromatography is newly designed and its performance is evaluated by the developed simulation tool

Investigation on Novel Magnetic Chromatography With Ferromagnetic Nano-Wires for Ion Separation

Magnetic chromatography is a very useful system for ion and/or fine magnetic particle separation due to strong magnetic field gradients in a very small flow channel. We have developed the magnetic chromatography system to separate the ions and fine particles. It is, however, difficult to separate the ions or the magnetic particles using the developed magnetic chromatography. It makes the strong magnetic field gradients in the flow channel, but the alternate direction of the gradient is not suitable for magnetic separation. Therefore, we have newly designed a novel magnetic column with ferromagnetic nano-wires. In this paper, the new magnetic column is presented, and the magnetic field and its gradients in the flow channel are investigated

Development of a Numerical Simulation Method for the Magnetic Separation of Magnetic Particles

Magnetic separation is a very useful tool in medicine manufacturing and sludge disposal and we have developed the magnetic chromatography system, which separates the magnetic particles or the ions from fluid due to its strong magnetic field gradients in the very small flow channel. There are many fine ferromagnetic wires on the wall of the developed magnetic column. A superconducting magnet applies a strong magnetic field to the magnetic column, and fine ferromagnetic wires make strong magnetic field gradients. It was, however, impossible to accurately evaluate the performance of the developed magnetic column due to the absence of a good simulation method. In order to enhance the accuracy of the simulation, it is necessary to couple the fluid dynamics simulation with the magnetic field simulation. Therefore, we have developed a simulation code dealing with the fluid dynamics, solving Navier-Stokes, control volume, and magnetic field equations simultaneously, and compared the simulation result with the experimental one to verify the validity

Magnetic Field and Fluid Flow Computation of Plural Kinds of Magnetic Particles for Magnetic Separation

Magnetic chromatography is an effective system for fine magnetic particle separation because of its strong magnetic field gradients. We have been developing the magnetic chromatography system to separate two or more kinds of fine magnetic particle with different sizes. To evaluate the performances of our developed system, we have also developed a simulation code with taking into account the fluid dynamics and magnetics. It is, however, difficult to consider two or more kinds of fine magnetic particles with different sizes in the simulation, because magnetic interferences between two or more kinds of magnetic particles need to be considered. Therefore, we have newly developed a simulation code to deal with magnetic interferences between two or more kinds of fine magnetic particles

Development of Numerical Analysis Method for Magnetic Separation of Magnetic Particle and Ion With Magnetic Chromatography

The magnetic chromatography is a very useful system for an ion and/or fine magnetic particle separation due to its strong magnetic field gradients in a very small flow channel. We have developed the magnetic chromatography system to separate the fine particles and ions. However, its numerical simulation is difficult, since the scale of the ions or fine particles in fluid is much different from the scale of the superconducting magnet generating the strong magnetic field. In order to accurately simulate the magnetic separation, it is necessary to develop the simulation code dealing with the multi-scale problem

Passive Shimming by Eliminating Spherical Harmonics Coefficients of all Magnetic Field Components Generated by Correction Iron Pieces

When a magnet generating highly homogeneous magnetic field is designed, a shimming is required. The shimming is usually performed so that the magnetic field only in the axis direction is compensated by iron pieces and/or coils. The compensation is commonly achieved by eliminating the coefficients of the spherical harmonics expansion of the magnetic field generated by the main magnet. Some papers showed the coefficients of the spherical harmonics expansion in only the z-direction for passive shimming. However, recently some magnets generate a magnetic field tilted from the z-axis, such as a magic-angle-spinning nuclear magnetic resonance/MRI. It is, therefore, necessary to eliminate the coefficients of the spherical harmonics expansion in the x- and y-direction