Minimality of rational knots C(2n+1,2m,2).

A nontrivial knot is called minimal if its knot group does not surject onto the knot groups of other nontrivial knots. In this paper, we determine the minimality of the rational knots C (2 n + 1 , 2 m , 2) in the Conway notation, where m ≠ 0 and n ≠ 0 , − 1 are integers. When | m | ≥ 2 , we show that the nonabelian SL 2 (ℂ) -character variety of C (2 n + 1 , 2 m , 2) is irreducible and therefore C (2 n + 1 , 2 m , 2) is a minimal knot. The proof of this result is an interesting application of Eisenstein's irreducibility criterion for polynomials over integral domains. [ABSTRACT FROM AUTHOR]