We study the sub-Riemannian structure determined by a left-invariant distribution of rank 2 on step 3 Carnot group dimension 5. prove conjectured cut times Yu. Sachkov for Cartan problem. Along proof, we obtain comparison with known in Engel group, and sufficient (generic) condition uniqueness length minimizer between two points. Hence reduce optimal synthesis to solving certain system equation...

In part I it was shown that for each k ≥ 1 the generalized Sato–Levine invariant detects a gap between k-quasi-isotopy of link and peripheral structure preserving isomorphism of the finest quotient Gk of its fundamental group, ‘functorially’ invariant under k-quasi-isotopy. Here we show that Cochran’s derived invariant β, provided k ≥ 3, and a series of μ̄-invariants, starting with μ̄(111112122) ...