  The magnetic and quadrupole moments of the $Z_{c}(4020)^+$, $Z_{c}(4050)^+$
and $Z_{c}(4600)^{+}$ states are calculated within the QCD light-cone sum
rules. The compact diquark-antidiquark interpolating currents and the
distribution amplitudes of the on-shell photon are used to extract the magnetic
and quadrupole moments of these states. The magnetic moments are acquired as
$\mu_{Z_{c}} = 0.50 ^{+0.22}_{-0.22}~\mu_N$, $\mu_{Z^{1}_{c}}=1.22
^{+0.34}_{-0.32}~\mu_N$, and $\mu_{Z^2_{c}}=2.40 ^{+0.53}_{-0.48}~\mu_N$ for
the $Z_{c}(4020)^+$, $Z_{c}(4050)^+$ and $Z_{c}(4600)^{+}$ states,
respectively. The magnetic moments evaluated for the $Z_{c}4020)^+$,
$Z_{c}(4050)^+$ and $Z_{c}(4600)^{+}$ states are sufficiently large to be
experimentally measurable. The magnetic moment is an excellent platform for
studying the internal structure of hadrons governed by the quark-gluon dynamics
of QCD because it is the leading-order response of a bound system to a weak
external magnetic field. The quadrupole moment results are
$\mathcal{D}_{Z_c}=(0.20 ^{+0.05}_{-0.04}) \times 10^{-3}~\mbox{fm}^2 $,
$\mathcal{D}_{Z_c^1}=(0.57 ^{+0.07}_{-0.08}) \times 10^{-3}~\mbox{fm}^2 $, and
$\mathcal{D}_{Z_c^2}=(0.30 ^{+0.05}_{-0.04}) \times 10^{-3}~\mbox{fm}^2 $ for
the $Z_{c}(4020)^+$, $Z_{c}(4050)^+$ and $Z_{c}(4600)^{+}$ states,
respectively. We obtain a non-zero, but small, value for the quadrupole moments
of the $Z_c$ states, which indicates a non-spherical charge distribution. The
nature and internal structure of these states can be elucidated by comparing
future experimental data on the magnetic and quadrupole moments of the
$Z_{c}(4020)^+$, $Z_{c}(4050)^+$, and $Z_{c}(4600)^{+}$ states with the results
of the present study.
