We mix together the Kontsevich integral, chord diagrams, Chinese characters, the Reshetikhin-Turaev knot invariants, the Poincare-Birkhoo-Witt theorem, the Harish-Chandra isomorphism, the Duuo isomorphism, the Kirillov formula and some minor Fourier-analysis computations. This enables us to make an intelligent guess for an exact formula for the Kontsevich integral of the unknot, and to conjectu...

Development on the design concept using OmniWheels to provide three degree-of-freedom (d.o.f) in a single spherical joint is presented in this paper. The purpose of this design is to provide the industrial robots that have limited movement flexibility with additional d.o.f in order to carry tasks that require complicated trajectories. In addition, this compact single joint will be installed as ...

We study the rational Kontsevich integral of torus knots. We construct explicitely a series of diagrams made of circles joined together in a tree-like fashion and colored by some special rational functions. We show that this series codes exactly the unwheeled rational Kontsevich integral of torus knots, and that it behaves very simply under branched coverings. Our proof is combinatorial. It use...

We design a vehicle with a steering system made of two independently rotatable wheels on the front. We quantify the effectiveness of the steering system in the mobility and maneuverability of the vehicle running in a box containing a layer ping-pong balls with a packing density 0.8, below the random close packing value 0.84 in 2D. The steering system can reduce the resistance exerted by the jam...