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tokens: example, Ganeshaiah Veena et al. ( 2021 ) have shown that massive haloes whose spin is perpendicular to the filament spine rotate on average faster (i.e. have higher spins) than haloes with spins along the filament spine. A similar effect has been pointed out by López et al. ( 2019 ), who have shown that the haloes which experienced the largest amount of spin growth are also more likely to have spin orientations perpendicular to the filaments surrounding them. 5.1 Net angular momentum growth of haloes Here, we follow the López et al. ( 2019 ) approach and define the net spin growth as the ratio between the present-day spin, , and the proto-halo spin, , in the initial conditions, which we take as the matter distribution at redshift z = 80. This definition is motivated by the TTT framework, in which the angular momentum at some time is the initial spin times a time-dependent growth factor (see equation 1 ). Thus, within the TTT approach, the net spin growth is basically the above growth factor. López et al. have shown that haloes with high ratios for their mass are more likely to have collapsed later and thus followed the TTT predictions for longer. This observation holds for the population as a whole, however not necessarily for individual haloes which could have a high ratio due to non-linear effects boosting their late-time spin growth. To keep the analysis simple, we follow the López et al. classification and for each narrow bin in mass we divide the halo population into three subsamples corresponding to the haloes with: highest, medium, and lowest net spin growth. We abbreviate these subsamples as H, M, and L, respectively. Each category corresponds to exactly a third of the haloes in each halo mass bin. This nomenclature is slightly different from the one used in López et al. ( 2019 ), where the same categories are referred as winner (W), median (M), and loser (L). Here, we decide to change the names of the samples (and the abbreviation of the one with the highest net spin growth, W→H) in order to better highlight their properties and the motivation of the classification. This analysis is illustrated in Fig. 8 , where the top panel show the net spin growth, , for individual haloes as a function of their mass. It shows that the ratio increases systematically with halo mass and illustrates why the H, M, and L subsamples need to be selected using narrow mass bins. The selection is further depicted in the bottom left-hand panel in Fig. 8 , which shows the distribution of net spin growth for a narrow mass bin. While most haloes have a narrow range in values, which for the given mass bin cluster around , the distribution has extended wings with a considerable fraction of haloes with ratios an order of magnitude higher or lower. The three subsamples, each containing a third of the haloes, are shown in the panel by the three shaded regions. Figure 8. Top panel: the halo net angular momentum growth (i.e. the present-day angular momentum divided by