  We study the conformal geometry of timelike curves in the (1+2)-Einstein
universe, the conformal compactification of Minkowski 3-space defined as the
quotient of the null cone of $\mathbb{R}^{2,3}$ by the action by positive
scalar multiplications. The purpose is to describe local and global conformal
invariants of timelike curves and to address the question of existence and
properties of closed trajectories for the conformal strain functional. Some
relations between the conformal geometry of timelike curves and the geometry of
knots and links in the 3-sphere are discussed.
