  Electromagnetic multipoles can be continuously mapped to tangent vectors on
the momentum sphere, the topology of which guarantees the existence of isolated
singularities. For pure (real or imaginary) vectors, those singularities
correspond to zeros of tangent fields, which can be classified by integer
Poincar\'{e} indices. Nevertheless, electromagnetic fields are generally
complex vectors, a comprehensive characterization of which requires the
introduction of line fields and line singularities categorized by half-integer
Hopf indices. Here we explore complex vectorial electromagnetic multipoles from
the perspective of line singularities, focusing on the special case of
polarization line field. Similar to the case of pure vectors, the
Poincar\'{e}-Hopf theorem forces the index sum of all line singularities to be
$2$, irrespective of the specific multipolar compositions. With this multipolar
insight, we further unveil the underlying structures of radiative
circularly-polarized Bloch modes of photonic crystal slabs, revealing their
topological origins with line singularities of Hopf indices. Our work has
established subtle connections between three seemingly unrelated but sweeping
physical entities (line singularities of Hopf indices, electromagnetic
multipoles, and Bloch modes), which can nourish new frames of visions and
applications fertilizing many related fields.
