  In this paper we provide a means to approximate Dirac operators with magnetic
fields supported on links in $\mathbb{S}^3$ (and $\mathbb{R}^3$) by Dirac
operators with smooth magnetic fields. We then proceed to prove that under
certain assumptions, the spectral flow of paths along these operators is the
same in both the smooth and the singular case. We recently characterized the
spectral flow of such paths in the singular case. This allows us to show the
existence of new smooth, compactly supported magnetic fields in $\mathbb{R}^3$
for which the associated Dirac operator has a non-trivial kernel. Using
Clifford analysis, we also obtain criteria on the magnetic link for the
non-existence of zero modes.
