It is well known that a Reeb vector field on $S^3$ has a periodic solution. Sharpening this result we shall show in this note that every Reeb vector field $X$ on $S^3$ has a periodic orbit which is unknotted and has self-linking number equal to $-1$. If the contact form $\lambda$ is non-degenerate, then there is even a periodic orbit $P$ which, in addition, has an index $\mu (P) \in \{2,3\}$, and which spans an embedded disc whose interior is transversal to $X$. The proofs are based on a theory for partial differential equations of Cauchy-Riemann type for maps from punctured Riemann surfaces into $\mathbb R \times S^3$, equipped with special almost complex structures related to the contact form $\lambda$ on $S^3$.