  Motivated by the study of ribbon knots we explore symmetric unions, a
beautiful construction introduced by Kinoshita and Terasaka in 1957. For
symmetric diagrams we develop a two-variable refinement $W_D(s,t)$ of the Jones
polynomial that is invariant under symmetric Reidemeister moves. Here the two
variables $s$ and $t$ are associated to the two types of crossings,
respectively on and off the symmetry axis. From sample calculations we deduce
that a ribbon knot can have essentially distinct symmetric union presentations
even if the partial knots are the same.
  If $D$ is a symmetric union diagram representing a ribbon knot $K$, then the
polynomial $W_D(s,t)$ nicely reflects the geometric properties of $K$. In
particular it elucidates the connection between the Jones polynomials of $K$
and its partial knots $K_\pm$: we obtain $W_D(t,t) = V_K(t)$ and $W_D(-1,t) =
V_{K_-}(t) \cdot V_{K_+}(t)$, which has the form of a symmetric product $f(t)
\cdot f(t^{-1})$ reminiscent of the Alexander polynomial of ribbon knots.
