\begin{document}$ Z_{c}(4020)^+ $\end{document}, \begin{document}$ Z_{c}(4050)^+ $\end{document}, and \begin{document}$ Z_{c}(4600)^{+} $\end{document} states are calculated within the QCD light-cone sum rules. The compact diquark-antidiquark interpolating currents and the distribution amplitudes of the on-shell photon are used to extract the magnetic and quadrupole moments of these states. The magnetic moments are acquired as \begin{document}$\mu_{Z_{c}}^{} = 0.50 ^{+0.22}_{-0.22}\; \mu_{N}^{}$\end{document}, \begin{document}$\mu_{Z^{1}_{c}}=1.22 ^{+0.34}_{-0.32}\; \mu_{N}^{}$\end{document}, and \begin{document}$\mu_{Z^2_{c}}=2.40 ^{+0.53}_{-0.48}\; \mu_{N}^{}$\end{document} for the \begin{document}$ Z_{c}(4020)^+ $\end{document}, \begin{document}$ Z_{c}(4050)^+ $\end{document}, and \begin{document}$ Z_{c}(4600)^{+} $\end{document} states, respectively. The magnetic moments evaluated for the \begin{document}$ Z_{c}4020)^+ $\end{document}, \begin{document}$ Z_{c}(4050)^+ $\end{document}, and \begin{document}$ Z_{c}(4600)^{+} $\end{document} states are sufficiently large to be experimentally measurable. The magnetic moment is an excellent platform for studying the internal structure of hadrons governed by the quark-gluon dynamics of QCD because it is the leading-order response of a bound system to a weak external magnetic field. The quadrupole moment results are \begin{document}$ \mathcal{D}_{Z_c}=(0.20 ^{+0.05}_{-0.04}) \times 10^{-3}\; \rm{fm}^2 $\end{document}, \begin{document}$ \mathcal{D}_{Z_c^1}=(0.57 ^{+0.07}_{-0.08}) \times 10^{-3}\; \rm{fm}^2 $\end{document}, and \begin{document}$ \mathcal{D}_{Z_c^2}=(0.30 ^{+0.05}_{-0.04}) \times 10^{-3}\; \rm{fm}^2 $\end{document} for the \begin{document}$ Z_{c}(4020)^+ $\end{document}, \begin{document}$ Z_{c}(4050)^+ $\end{document}, and \begin{document}$ Z_{c}(4600)^{+} $\end{document} states, respectively. We obtain a non-zero, but small, value for the quadrupole moments of the \begin{document}$ Z_c $\end{document} states, which indicates a non-spherical charge distribution. The nature and internal structure of these states can be elucidated by comparing future experimental data on the magnetic and quadrupole moments of the \begin{document}$ Z_{c}(4020)^+ $\end{document}, \begin{document}$ Z_{c}(4050)^+ $\end{document}, and \begin{document}$ Z_{c}(4600)^{+} $\end{document} states with the results of the present study."> Magnetic and quadrupole moments of the <inline-formula><tex-math id="M1">\begin{document}${\boldsymbol Z_{\boldsymbol c}\bf (4020)^+} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/aa479384-8533-4d27-a94f-edd8f1e5ab52/CPC-2023-0308_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/aa479384-8533-4d27-a94f-edd8f1e5ab52/CPC-2023-0308_M1.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ {\boldsymbol Z_{\boldsymbol c}\bf (4050)^+} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/aa479384-8533-4d27-a94f-edd8f1e5ab52/CPC-2023-0308_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/aa479384-8533-4d27-a94f-edd8f1e5ab52/CPC-2023-0308_M2.png"/></alternatives></inline-formula>, and <inline-formula><tex-math id="M3">\begin{document}$ {\boldsymbol Z_{\boldsymbol c}\bf (4600)^{+}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/aa479384-8533-4d27-a94f-edd8f1e5ab52/CPC-2023-0308_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/aa479384-8533-4d27-a94f-edd8f1e5ab52/CPC-2023-0308_M3.png"/></alternatives></inline-formula> states in the diquark-antidiquark picture -
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