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.

Torus-covering knot groups and their irreducible metabelian $SU(2)$-representations

“…The same statement with Theorem 1.1 holds true for irreducible metabelian SL(2, C)-representations of the knot group of a classical knot, due to Nagasato [14]. And Fukuda [6] studied irreducible metabelian SL(2, C)-representations of the knot group of a certain surface-knot called a branched twist spin, which is determined from a classical knot and a pair of integers.…”

“…We remark that in this paper, we focus on SU (2)-representations, but, essentially by the same argument, our results hold true also for irreducible metabelian SL(2, C)-representations. See the argument in [6,14,15]. The paper is organized as follows.…”