Representations of branched twist spins with a non-trivial center of order 2

TL;DR

This paper investigates the algebraic representations of branched twist spins, focusing on ${ m SL}_2(oldsymbol{ ext{Z}}_3)$-representations and dihedral group representations, providing conditions for existence and counting specific representations.

Contribution

It introduces new criteria for ${ m SL}_2(oldsymbol{ ext{Z}}_3)$-representations and counts dihedral group representations for branched twist spins, advancing understanding of their algebraic structures.

Findings

Sufficient condition for ${ m SL}_2(oldsymbol{ ext{Z}}_3)$-representation existence.

Number of even-ordered dihedral group representations determined.

Enhanced understanding of the algebraic properties of branched twist spins.

Abstract

It is known that a presentation of the knot group of a branched twist spin is obtained from a Wirtinger presentation of the original 1-knot group by adding a generator corresponding to a regular orbit of the circle action and a certain relator. In particular, the additional generator is an element of the center of the knot group. In this paper, we focus on ${\rm SL}_2(\mathbb{Z}_3)$-representations and dihedral group representations. For the former case, we give a sufficient condition for the existence of an ${\rm SL}_2(\mathbb{Z}_3)$-representation for a branched twist spin. For the latter case, we determine the number of even-ordered dihedral group representations of branched twist spins.