Irreducible SL(2, ℂ)-metabelian representations of branched twist spins

Abstract

An [Formula: see text]-branched twist spin is a fibered [Formula: see text]-knot in [Formula: see text] which is determined by a [Formula: see text]-knot [Formula: see text] and coprime integers [Formula: see text] and [Formula: see text]. For a [Formula: see text]-knot, Nagasato proved that the number of conjugacy classes of irreducible [Formula: see text]-metabelian representations of the knot group of a [Formula: see text]-knot is determined by the knot determinant of the [Formula: see text]-knot. In this paper, we prove that the number of irreducible [Formula: see text]-metabelian representations of the knot group of an [Formula: see text]-branched twist spin is determined up to conjugation by the determinant of the associated [Formula: see text]-knot in the orbit space by comparing a presentation of the knot group of the branched twist spin with the Lin’s presentation of the knot group of the [Formula: see text]-knot.