Signatures of TQFTs and trace fields of two-bridge knots

Let $0<s<r$ be coprime odd integers. We show that the Frobenius algebras governing the signatures of SO${}_3$ TQFTs at the root $q=\exp (i\pi s\mathrm{/}r)$ 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.