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

Abstract

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 [Formula: see text] in the Conway notation, where [Formula: see text] and [Formula: see text] are integers. When [Formula: see text], we show that the nonabelian [Formula: see text]-character variety of [Formula: see text] is irreducible and therefore [Formula: see text] is a minimal knot. The proof of this result is an interesting application of Eisenstein’s irreducibility criterion for polynomials over integral domains.