Some non-trivial PL knots whose complements are homotopy circles

We show that there exist non-trivial piecewise-linear (PL) knots with isolated singularities Sn−2 ⊂ Sn, n ≥ 5, whose complements have the homotopy type of a circle. This is in contrast to the case of smooth, PL locally-flat, and topological locally-flat knots, for which it is known that if the complement has the homotopy type of a circle, then the knot is trivial. It is well-known that if the complement of a smooth, piecewise linear (PL) locally-flat, or topological locally-flat knot K ⊂ S, K ∼= Sn−2, n ≥ 5, has the homotopy type of a circle, then K is equivalent to the standard unknot in the appropriate category (see Stallings [11] for the topological case and Levine [6] and [8, §23] for the smooth and PL cases). This is also true of classical knots S ↪→ S (see [10, §4.B]), for which these categories are all equivalent, and in the topological category for locally-flat knots S ↪→ S by Freedman [2, Theorem 6]. By contrast, Freedman and Quinn showed in [3, §11.7] that any classical knot with Alexander polynomial 1 bounds a topological locally-flat D in D whose complement is a homotopy circle, and by collapsing the boundary, one