Are there any restrictions on $\pi\_1(S^3\backslash K)$ for a (tame) knot $K$ besides having $\pi\_1^{\text{ab}}(S^3\backslash K)=\mathbb{Z}$?

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So we have 1) finite presentation, 2) $H\_1=\mathbb{Z}$, 3) $H\_2=0$, and I saw somewhere 4) the knot group must be the normal closure of a single element.

Apparently this characterizes knot groups in dimensions $n\geq5$ (i.e. $\pi\_1(S^n\backslash S^{n-2})$), but fails to do so in lower dimensions. Basically I just wanted to know if there is simple characterization of knot groups (not "it has a presentation that looks like a wirtinger presentation")