A mathematical model is developed for a mobile robot with three ball-wheels. The resulting model turns out to be a Chaplygin system, which allows the dynamics equations and the kinematic constraints to be integrated separately. The system is driven by six actuators: one actuator connected to one roller of each wheel, and three more actuators controlling the orientation of the ball-carriers with...

By considering spaces of directed Jacobi diagrams, we construct a diagrammatic version of the Etingof-Kazhdan quantization of complex semisimple Lie algebras. This diagrammatic quantization is used to provide a construction of a directed version of the Kontsevich integral, denoted ZEK, in a way which is analogous to the construction of the Reshetikhin-Turaev invariants from the R-matrices of th...

Irma Nykänen1,2,3* Tiina H Rissanen4 and Sirpa Hartikainen1,3 1Kuopio Research Centre of Geriatric Care, University of Eastern Finland, Kuopio, Finland 2Research Centre for Comparative Effectiveness and Patient Safety (RECEPS), University of Eastern Finland, Kuopio, Finland 3School of Pharmacy, Faculty of Health Sciences, University of Eastern Finland, Kuopio, Finland 4Institute of Public Healt...

We conjecture an exact formula for the Kontsevich integral of the unknot, and also conjecture a formula (also conjectured independently by Deligne [De]) for the relation between the two natural products on the space of uni-trivalent diagrams. The two formulas use the related notions of “Wheels” and “Wheeling”. We prove these formulas ‘on the level of Lie algebras’ using standard techniques from...

Using elementary equalities between various cables of the unknot and the Hopf link, we prove the Wheels and Wheeling conjectures of [5, 9], which give, respectively, the exact Kontsevich integral of the unknot and a map intertwining two natural products on a space of diagrams. It turns out that the Wheeling map is given by the Kontsevich integral of a cut Hopf link (a bead on a wire), and its i...

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...

In this paper, the chromaticity of K3-gluings of two wheels is studied. For each even integer n ≥ 6 and each odd integer 3 ≤ q ≤ [n/2] all K3-gluings of wheels Wq+2 and Wn−q+2 create an χ-equivalent class.