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

Abstract

A branched twist spin is a 2-knot consisting of exceptional orbits and fixed points of a circle action on the four sphere. It is a generalization of the twist spun knot, and its knot group 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 this paper, we study on ${\mathrm{SL}}_2\left(Z_3\right)$-representations and dihedral group representations. For the former case, we give a sufficient condition for the existence of an ${\mathrm{SL}}_2\left(Z_3\right)$-representation for a branched twist spin. For the latter case, we determine the number of 4k-ordered dihedral group representations of branched twist spins. As an application, we can show non-equivalence between two branched twist spins by counting dihedral representations of their knot groups.