Whitehead doubling, rank estimate and nonembeddability of contractible open manifolds

Let $K$ be a nontrivial knot. For each $n\in \mathbb{N}$, we prove that the rank of its $n$th iterated Whitehead doubled knot group ${\pi }_1(S^3\setminus {\mathrm{WD}}^n(K))$ is bounded below by $n+1$. As an application, we show that there exist infinitely many non-homeomorphic contractible open $n$-manifolds ($n\ge 3$) which cannot embed in a compact, locally connected and locally 1-connected $n$-dimensional metric space.