Some families of minimal elements for a partial ordering on prime knots

Abstract

We show that all twist knots and certain double twist knots are minimal elements for a partial ordering on the set of prime knots. The keys 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.