Let \( B^n \) denote the smooth \( n \)-ball, and \( g:B^n\to B^{n+2} \) be a smooth proper embedding, that is, \( g^{-1}(\partial B^{n+2}) = \partial B^n \). Then \( (B^{n+2},gB^n) \) is a smooth ball pair, possibly knotted. By taking the cartesian product of \( (B^{n+2},gB^n) \) with an interval \( I \), we obtain the ball pair
\[ (B^{n+3},g^{\prime} B^{n+1}) = (B^{n+2},gB^n)\times I ,\]
with \( g^{\prime} = g\times id \). In this paper we construct, for all \( n\geq 2 \), infinitely many distinct examples of smooth knotted ball pairs whose product with an interval yields the unknotted ball pair. This improves the result of Kato [1969], who produced such examples for \( n\geq 4 \). The examples obtained here are in the lowest possible dimension (\( n = 2 \)), because if \( (B^3,gB^1)\times I \) is unknotted, then \( (B^3, gB^1) \) must itself be unknotted, because the complement is a homotopy circle.
We give two different methods for constructing examples in the lowest-dimensional situation \( (B^4,gB^2) \), and produce the higher-dimensional examples by \( P \)-spinning each of the \( (B^4,gB^2) \) about an unknotted face. The first of the methods is a simple handlebody construction, which is really the analogue for pairs of the Mazur manifold phenomenon [Zeeman 1962]. The second method is derived from a more general unknotting result, a corollary of which is the following:
The untwisted double of any slice knot bounds a smooth ball pair \( (B^4,gB^2) \) such that \( (B^4,gB^2)\times I \) is unknotted.
