ering
\subfigure[]{\label{fig:vort-C}\includegraphics[width=0.32 \textwidth]{VCh11.png}}\hspace{0.3cm}
\subfigure[]{\label{fig:vort-Y}\includegraphics[width=0.32 \textwidth]{VY11.png}}\\
\subfigure[]{\label{fig:vort-X}\includegraphics[width=0.32 \textwidth]{VX11.png}}\hspace{0.3cm}
\subfigure[]{\label{fig:vort-R}\includegraphics[width=0.32 \textwidth]{VR11.png}}
	\caption{Histograms of relative vortex size and typical snapshots showing the direction of the curl at the position of each particle, indicated by two-coloured light and dark hemispheres, corresponding to $\mu^2=5$. (a) For chain-like SMPs; (b) Y-like SMPs; (c) X-like SMPs; (d) ring-like SMPs.}
	\label{fig:vort-hist}
\end{figure*}
Moreover, there are three important tendencies in the behaviour of chain-, Y- and X-like SMPs to be noticed in Table \ref{table:in-susc}. First, comparing the values of $\kappa$ for different $\mu^2$, one can see that the decrease of the susceptibility due to the formation of compact clusters is much larger for strongly interacting particles.  Besides that, for $\mu^2=2$, any crosslinking leads to an increase in susceptibility. This is related to the fact that in the system of noncrosslinked dipolar soft spheres with $\rho=0.05$ and $\mu^2 = 2$, the particles are rather weakly correlated, as it can be seen from the value of the initial susceptibility which is basically equal to that of Langevin. If, instead, $\mu^2=5$, crosslinking in X-like SMPs leads to a slight decrease in the initial susceptibility: each linear segment of an X-like SMP made of 9 magnetic particles has only two beads. The average chain length in the system of DSS, see for example \cite{wang_pre}, is around 2.5. Thus, in this case, the crosslinking weakens the correlations. Clearly, the strongest enhancement of the interparticle correlations is observed for chain-like topology. At the same time, for the latter, the value of $\kappa$ turns out to be the largest.

In order to explain this, we thoroughly analysed the distribution of dipoles inside the clusters. Such distribution defines a discrete vector field, $F$ (not to be confused with the magnetic field created by the distribution of dipoles). $F$ is given by the position of the centre of each particle and the vector of its magnetic moment. By treating the distribution of dipole moments as a vector field one can calculate its curl. The latter can show if there are vortex-like arrangements of dipoles and if the number and size distribution of these vortices depends on the constituent SMP topology. This is important because vortex-like structures, possesing a low net magnetic moment, can be responsible for drastic decrease of the susceptibility. We calculated curls for each cluster found in simulations on a grid provided by particle centres using the weighted regression method due to its robustness \cite{2003-petrovskaya,2011-correa}. The calculations were performed for all clusters whose size was in a range between 45 and 48 SMPs. As 3D vector plots are very difficult to visualise, we processed the curl vectors further: for each curl vector we found the neighbourhood (neighbouring particles) in which the orientation of the curls of two adjacent particles deviate by less than 15\% ($\sim \pi/6$). If the neighbourhood contained less than four particles it was discarded, otherwise it was named as a vortex of a given size. Additionally, the vortex size was normalised by the total number of particles in the cluster. With this procedure, we calculated the histograms of relative vortex sizes for solutions of different SMPs. The results for  $\mu^2=5$ are presented in Fig.~\ref{fig:vort-hist}. It is clearly seen that for chain-like SMP clusters, the size distribution of vortices is bimodal (Fig. \ref{fig:vort-C}). For each distribution, we also include a typical cluster with particles showing two different light/dark coloured hemispheres, so that the division plane is perpendicular to the curl in this point. Visual inspection of the cluster snapshot in the inset of Fig. \ref{fig:vort-C} reveals the presence of a large neighbourhood of particles turned with their dark side to the reader. The curl vectors calculated in the centres of these particles are almost coaligned, thus, these particles are forming a large vortex. In none of the other three Figs. \ref{fig:vort-Y} -- \ref{fig:vort-R} one finds such large vortices, however, there are many smaller local ones. For clusters made of ring-like SMPs, some of the vortices coincide with the rings themselves, as it can be seen in the inset of Fig. \ref{fig:vort-R}. The absence of large vortices in the clusters of Y-, X- and ring-like SMPs has a simple geometrical explanation: in Fig. \ref{fig:shapes} (c) -- (e) the dipoles are shown in the crosslinked orientation and any deviation from this orientation is penalised by elastic forces, thus making their rotational degrees of freedom on average more constrained than for chain-like SMPs (Fig. \ref{fig:shapes} (b)).   

\section{Conclusions}
In the present study we analysed zero-field magnetic properties of suspensions composed of Stockmayer SMPs of four different topologies. We found that, due to the formation of compact clusters, the overall initial magnetic susceptibility decreases dramatically for chain-, Y- and X-like topologies. However, this decrease for chain-like SMPs is much stronger than that of other topologies, especially for systems with strong magnetic interactions. This can be explained by calculating the curls of magnetic moment distributions. It turned out that chain-like SMPs, when self-assembling in large compact clusters, rearrange their particle dipole moments inside the clusters to form large vortexes, whose total magnetic moment is negligible. Other SMPs cannot exhibit the same behaviour due to the constraints on the particle rotations away from the crosslinked backbone. This result might be of importance for the behaviour of Stockmayer SMP clusters in flow under the applied magnetic fields, particularly if one aims, for instance, at optimising transport properties.

\section{Acknowledgements}
Research supported by the Russian Science Foundation Grant No.19-72-10033. S.S.K. acknowledges support from the Austrian Research Fund (FWF), START-Projekt Y 627-N27.


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