  Twisted light is light carrying orbital angular momentum. The profile of such
a beam is a ring-like structure with a node at the beam axis, where a phase
singularity exits. Due to the strong spatial inhomogeneity the mathematical
description of twisted-light--matter interaction is non-trivial, in particular
close to the phase singularity, where the commonly used dipole-moment
approximation cannot be applied. In this paper we show that, if the
polarization and the orbital angular momentum of the twisted-light beam have
the same sign, a Hamiltonian similar to the dipole-moment approximation can be
derived. However, if the signs of polarization and orbital angular momentum
differ, in general the magnetic parts of the light beam become of significant
importance and an interaction Hamiltonian which only accounts for electric
fields, as in the dipole-moment approximation, is inappropriate. We discuss the
consequences of these findings for twisted-light excitation of a semiconductor
nanostructures, e.g., a quantum dot, placed at the phase singularity.
