On the Total CR Twist of Transversal Curves in the 3-Sphere

We investigate the total CR twist functional on transversal curves in the standard CR 3-sphere ${\mathrm{S}}^3\subset {\mathbb{C}}^2$. The question of the integration by quadratures of the critical curves and the problem of existence and properties of closed critical curves are addressed. A procedure for the explicit integration of general critical curves is provided and a characterization of closed curves within a specific class of general critical curves is given. Experimental evidence of the existence of infinite countably many closed critical curves is provided.