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

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 SL2(Z3)-representations and dihedral group representations. For the former case, we give a sufficient condition for the existence of an SL2(Z3)-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. (c) 2025 Published by Elsevier B.V.