  We show that all twist knots, certain double twist knots and some other
2-bridge knots are minimal elements for the partial ordering on the set of
prime knots. The key to these results are presentations of their character
varieties using Chebyshev polynomials and a criterion for irreducibility of a
polynomial of two variables. These give us an elementary method to discuss the
number of irreducible components of the character varieties, which concludes
the result essentially.
