On the Cauchy-Riemann Geometry of Transversal Curves in the 3-Sphere

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