Given a knot K and a representation one can define the twisted Alexander polynomials of the knot K. Degrees of twisted Alexander polynomials give in particular lower bounds on the Seifert genus. Using recent results of Agol and Wise we will show that twisted Alexander polynomials detect the genus of hyperbolic knots. This result has an application to the problem of determing the minimal complexity of surfaces in 4-manifolds which are S^1-bundles over a 3-manifold.