On minimal elements for a partial order of prime knots

In this paper, applying Chebyshev polynomials we give a basic proof of the irreducibility over the complex number field of the defining polynomial of SL 2 ( C ) -character variety of twist knots in infinitely many cases. The irreducibility, combined with a result in the paper of M. Boileau, S. Boyer, A.W. Reid and S. Wang in 2010, shows the minimality of infinitely many twist knots for a partial order on the set of prime knots defined by using surjective group homomorphisms between knot groups. In Appendix B, we also give a straightforward proof of the result of Boileau et al.