  It is still unknown whether there is a nontrivial knot with Jones polynomial
equal to that of the unknot. Tanaka shows that if an amphichiral knot is a
symmetric union of the unknot with one twist region, then its Jones polynomial
is trivial. Hence, he proposes that if any of these knots were nontrivial, a
nontrivial knot with trivial Jones polynomial would exist. We first show such a
knot is always trivial and hence cannot be used to answer the above question.
We then generalize the argument to symmetric unions of any knots and show that
if a symmetric union of a knot $J$ with one twist region is amphichiral, then
it is the connected sum of $J$ and its mirror image $-J$.
