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

Fumikazu Nagasato, Anh T. Tran

Published 2013-01-01, updated 2014-09-02Version 2

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.