We improve the best known lower bounds on the distance between two points of a Morse cluster in R, with ρ ∈ [4.967, 15]. Our method is a generalization of the one applied to the Lennard-Jones potential in [2], and it also leads to improvements of lower bounds for the energy of a Morse cluster. Some of the numerical results have been announced in [1].

We study the new formulas of Th. Fiedler for the degree-3-Vassiliev invariants for knots in the 3-sphere and solid torus and present some results obtained by them. We show that a knot with Jones polynomial consisting of exactly two monomials must have at least 20 crossings.