A geometric formula for Haefliger knots

AbstractHaefliger has shown that a smooth embedding of the (4k−1)-sphere in the 6k-sphere can be knotted in the smooth sense. In this paper, we give a formula with which we can detect the isotopy class of such a Haefliger knot. The formula is expressed in terms of the geometric characteristics of an extension, analogous to a Seifert surface, of the given embedding. In particular, the Hopf invariant associated to the extension plays a crucial role. This leads us to a new characterisation of Haefliger knots

High-codimensional knots spun about manifolds

Using spinning we analyze in a geometric way Haefliger's smoothly knotted (4k-1)-spheres in the 6k-sphere. Consider the 2-torus standardly embedded in the 3-sphere, which is further standardly embedded in the 6-sphere. At each point of the 2-torus we have the normal disk pair: a 4-dimensional disk and a 1-dimensional proper sub-disk. We consider an isotopy (deformation) of the normal 1-disk inside the normal 4-disk, by using a map from the 2-torus to the space of long knots in 4-space, first considered by Budney. We use this isotopy in a construction called spinning about a submanifold introduced by the first-named author. Our main observation is that the resultant spun knot provides a generator of the Haefliger knot group of knotted 3-spheres in the 6-sphere. Our argument uses an explicit construction of a Seifert surface for the spun knot and works also for higher-dimensional Haefliger knots.Comment: 14 pages, 10 figure

COMPOSITIONS OF EQUI-DIMENSIONAL FOLD MAPS

Abstract. According to Ando's theorem, the oriented bordism group of fold maps of n-manifolds into n-space is isomorphic to the stable n-stem. Among such fold maps we define two geometric operations corresponding to the composition and to the Toda bracket in the stable stem through Ando's isomorphism. By using the operations we explicitly construct several fold maps with favourable properties, including a fold map which represents the generator of the stable 6-stem

Measurement of Excited States of Sb Impurity in Si by Traveling-Wave Method

The ground and excited states of Sb atom in Si, 1s (A 1), 1s (T 2), 1s (E), and 2p0, were measured by using a traveling-wave method. The Sb-doped Si crystal with donor concentration of 2 × 1015 cm−3 was placed the distance of 5 μm above a piezoelectric crystal in the fringe field of a surface acoustic wave. The free electrons excited from the bound states of the Sb atom are drifted by the traveling-wave, and thus lose their energy as the Joule heat through lattice and ion scattering processes. A strong temperature-dependent energy loss of the traveling-wave can be observed at temperatures below 200 K. The values of the bound states of the Sb atom can be characterized by using the Arrhenius plot for thermal activation process of the electrons in the bound states. The measurements were carried out at two frequencies of the traveling-wave, 50 MHz and 200 MHz. At the frequency of 50 MHz, the dielectric properties of the Si crystal are governed by dopant polarization but by electronic polarization at 200 MHz. We found that measurement accuracy of the bound states depends mainly on the electron mobility and the dielectric constant of the Si crystal, which are sensitive to the frequency and strength of the traveling-wave as well as electronic polarization properties of the Si crystal

Diffraction profile analysis of olivines in thin sections of meteorites by the micro-region Laue method using synchrotron radiation

The lattice distortion of minerals included in a thin section of meteorites was measured by a newly developed X-ray technique for microregion analysis (8μm diameter) using synchrotron radiation. This technique using the Laue method with an imaging plate (IP) as a two-dimensional detector, as well as a micro-pinhole, was applied to olivine grains in two L6 chondrites (Y-7304 and Y-7305). The orientation matrix calculated from one end of the full width at half maximum (FWHM) along the elongated diffraction profiles is different from that based on the other end of the FWHM. Rotation about a certain axis equates one matrix to the other. The rotation angles for the Y-7304 and Y-7305 olivines were calculated to be 0.83 and 0.15 degrees, respectively, and are compatible with the FWHM of diffraction profiles. The results of the present X-ray study correspond quantitatively to the wavy extinction observed through an optical microscope. This method will be applied to analyze the lattice distortion of minerals caused by shock effects in meteorites

振動リード法によるBi2Sr2Ca2Cu3Ox超電導テープの内部摩擦測定

The 33rd Symposium on UltraSonic Electronics (USE2012): 第33回超音波エレクトロニクスの基礎と応用に関するシンポジウム, 2012年11月13日～15日, 千葉大学, 千