On the Cauchy-Riemann geometry of transversal curves in the 3-sphere

Let S^3 be the unit sphere of C^2 with its standard Cauchy-Riemann (CR) structure. This paper investigates the CR geometry of curves in S^3 which are transversal to the contact distribution, using the local CR invariants of S^3. More specifically, the focus is on the CR geometry of transversal knots. Four global invariants of transversal knots are considered: the phase anomaly, the CR spin, the Maslov index, and the CR self-linking number. The interplay between these invariants and the Bennequin number of a knot are discussed. Next, the simplest CR invariant variational problem for generic transversal curves is considered and its closed critical curves are studied.