Fundamental Quandles Do Not Determine the First Postnikov Invariant of 2-Knots

Abstract

For every admissible pair of Brieskorn parameters in the Plotnick--Suciu construction, we obtain a pair of oriented $2$-knots whose knot groups are isomorphic, whose second homotopy modules are semilinearly isomorphic under a suitable group isomorphism, and whose fundamental quandles are isomorphic, while no compatible group and module isomorphisms carry one first Postnikov invariant to the other. Consequently, their exteriors are not homotopy equivalent. Thus, the knot group, the second homotopy module up to semilinear equivalence, and the fundamental quandle do not determine the homotopy type of an oriented $2$-knot exterior. Using an alternative construction from Suciu's thesis based on punctured lens spaces, the same peripheral argument yields, for every $N\geq2$, a family of $N$ oriented $2$-knots with these properties. We additionally show that the Tanaka--Taniguchi examples with isomorphic knot groups and distinct fundamental quandles have pairwise inequivalent second homotopy modules: no isomorphism between two of the knot groups makes the corresponding second homotopy modules semilinearly isomorphic.