Non abelian Reidemeister torsion and volume form on the SU(2)-representation space of knot groups

For a knot K in $S^3$ and a regular representation $\rho$ of its group $G_K$ into SU(2) we construct a non abelian Reidemeister torsion on the first twisted cohomology group of the knot exterior. This non abelian Reidemeister torsion provides a volume form on the SU(2)-representation space of $G_K$. In another way, we construct according to Casson--or more precisely taking into account Lin's and Heusener's further works--a volume form on the SU(2)-representation space of $G_K$. Next, we compare these two apparently different points of view--the first by means of the Reidemeister torsion and the second defined ``a la Casson"--and finally prove that they define the same topological knot invariant.Comment: 36 pages, 2 figures. to appear in Ann. Institut Fourie

On the asymptotic expansion of the colored Jones polynomial for torus knots

In the asymptotic expansion of the hyperbolic specification of the colored Jones polynomial of torus knots, we identify different geometric contributions, in particular Chern--Simons invaraint and Reidemeister torsion.Comment: 24 pages, 1 figure. to appear in Math. An

The L^2-Alexander torsion of 3-manifolds

We introduce $L^2$-Alexander torsions for 3-manifolds, which can be viewed as a generalization of the $L^2$-Alexander polynomial of Li--Zhang. We state the $L^2$-Alexander torsions for graph manifolds and we partially compute them for fibered manifolds. We furthermore show that given any irreducible 3-manifold there exists a coefficient system such that the corresponding $L^2$-torsion detects the Thurston norm.Comment: 47 pages v3: fixed many typos, updated references and improved the exposition, following the referees suggestion

A Euclidean Geometric Invariant of Framed (Un)Knots in Manifolds

We present an invariant of a three-dimensional manifold with a framed knot in it based on the Reidemeister torsion of an acyclic complex of Euclidean geometric origin. To show its nontriviality, we calculate the invariant for some framed (un)knots in lens spaces. Our invariant is related to a finite-dimensional fermionic topological quantum field theory

On the asymptotic expansion of the colored Jones polynomial for torus knots

In the asymptotic expansion of the hyperbolic specification of the colored Jones polynomial of torus knots, we identify different geometric contributions, in particular Chern-Simons invariant and twisted Reidemeister torsion with coefficients in the adjoint representatio

A Programmable Vision Chip with High Speed Image Processing

International audienceA high speed Analog VLSI Image acquisition and pre-processing system is described in this paper. A 64×64 pixel retina is used to extract the magnitude and direction of spatial gradients from images. So, the sensor implements some low-level image processing in a massively parallel strategy in each pixel of the sensor. Spatial gradients, various convolutions as Sobel filter or Laplacian are described and implemented on the circuit. The retina implements in a massively parallel way, at pixel level, some various treatments based on a four-quadrants multipliers architecture. Each pixel includes a photodiode, an amplifier, two storage capacitors and an analog arithmetic unit. A maximal output frame rate of about 10000 frames per second with only image acquisition and 2000 to 5000 frames per second with image processing is achieved in a 0.35 μm standard CMOS process. The retina provides address-event coded output on three asynchronous buses, one output is dedicated to the gradient and both other to the pixel values. A prototype based on this principle, has been designed. Simulation results from Mentor GraphicsTMsoftware and AustriaMicrosystem Design kit are presented