# Irreducible $SL(2,\mathbb{C})$-metabelian representations of branched twist spins

## TL;DR

This paper extends Lin's result on irreducible $SL(2,b{C})$-metabelian representations from knots to branched twist spins, showing their count depends on the original knot's determinant.

## Contribution

It proves that the number of such representations for branched twist spins is determined by the determinant of the original knot, generalizing Lin's result.

## Key findings

- Number of irreducible $SL(2,b{C})$-metabelian representations depends on the knot determinant.
- Comparison of group presentations links the representations of branched twist spins to the original knot.
- The result applies to $(m,n)$-branched twist spins, broadening understanding of their algebraic properties.

## Abstract

An $(m,n)$-branched twist spin is a fibered $2$-knot in $S^4$ which is determined by a $1$-knot $K$ and coprime integers $m$ and $n$. For a $1$-knot, Lin proved that the number of irreducible $SL(2,\mathbb{C})$-metabelian representations of the knot group of a $1$-knot up to conjugation is determined by the knot determinant of the $1$-knot. In this paper, we prove that the number of irreducible $SL(2,\mathbb{C})$-metabelian representations of the knot group of an $(m,n)$-branched twist spin up to conjugation is determined by the determinant of a $1$-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 $1$-knot.