  Let $0<s<r$ be coprime odd integers. We show that the Frobenius algebras
governing the signatures of SO$_3$ TQFTs at the root $q=\exp(i\pi s /r)$
contain (and are often equal to) the trace field of the two-bridge knot of
parameters $(r,s)$. This gives an intriguing relationship between these two a
priori unrelated objects of low-dimensional topology.
