2-knots with the same knot group but different knot quandles

Abstract

We give a first example of a pair of 2-knots that share the same knot group but have different knot quandles. In fact, we give infinitely many triples of twist spins that share the same knot group but have mutually different knot quandles. To this end, we prove that the type of the knot quandle of an 𝑛-twist spin of a non-trivial knot is equal to 𝑛. By using the latter result, we also complete the classification of the twist spins with finite knot quandles by distinguishing the 2-knots in Inoue's list.