  A theoretical analysis is presented for the excitation of single
many-electron atoms and ions by twisted (or vortex) light. Special emphasis is
put on excitations that can proceed via several electric and magnetic multipole
channels. We argue that the relative strength of these multipoles is very
sensitive to the topological charge and kinematic parameters of the incident
light and can be strongly modified with respect to the plane-wave case. Most
remarkably, the modification of multipole transitions by twisted radiation can
be described by means of a geometrical factor. This factor is independent of
the shell structure of a particular target atom and just reflects the
properties of the light beam as well as the position of an atom with respect to
the vortex axis. An analytical expression for the geometrical factor is derived
for Bessel photons and for a realistic experimental situation in which the
position of an atom is not well determined. To illustrate the use of the
geometrical factor for the analysis of (future) measurements, detailed
calculations are presented for the $3s3p \, ^3P_1 \to 3s 3p \, ^1P_1$
excitation in neutral Mg.
