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A branched twist spin is a generalization of twist spun knots, which appearedin the study of locally smooth circle actions on the $4$-sphere due toMontgomery, Yang, Fintushel and Pao. In this paper, we give a sufficientcondition to distinguish non-equivalent, non-trivial branched twist spins byusing knot determinants. To prove the assertion, we give a presentation of thefundamental group of the complement of a branched twist spin, which generalizesa presentation of Plotnick, calculate the first elementary ideals and obtainthe condition of the knot determinants by substituting $-1$ for theindeterminate.

Branched twist spins and knot determinants