## 3 Results  
We benchmarked the scTSA method on the zebrafish single-cell RNA sequencing data with 38,731 cells and 25 cell types over 12 time steps [23]. The dataset studied the embryogenesis, which is the process where the cells gradually differentiate into distinct fates through stages of transcriptional change. The goal of this study was to facilitate a comprehensive identification of cell types with their time stamps in order to reconstruct their developmental trajectories (e.g., transcriptional states, branch points, and asynchrony). As the gene expression profiles obtained from the vertebrate embryo were time stamped from 3.3 to 12 hours post-fertilization (hpf), they provided a perfect testbed for time series analysis to reconstruct the transcriptional trajectories and characterize the time-dependent development properties. We processed the scRNA sequencing data into entries of 103 dimensions corresponding to the expression levels of 103 significant genes (which we selected using scTDA). We then standardized the features by removing the mean and scaling to the unit variance. Before we performed the persistent homology, we first embedded the dataset into a low-dimensional space using dimension reduction. In Figure 5, we embedded the data using principal component analysis (PCA), t-distributed stochastic neighbor embedding (TSNE) [38], and uniform manifold approximation and projection (UMAP) [39] and colored them by time step. We observed that they all demonstrated a temporal gradient. We performed persistent homology and temporal persistent homology on the scaled dataset. In this case, \(\tau\) was set to be one (meaning that we only cared about the linkage formed between consecutive time points). Figure 6 compares the persistence diagrams for the two approaches. From the persistence diagrams, the persistent features detected by persistent homology were not noticeably different from those detected by temporal persistent homology. While not a focus of this work, further study using downstream machine learning tasks can potentially pinpoint the benefits of these temporal persistent features. For the simplicial analysis, we first grouped the data by their time steps. The data collected at the 12 time steps were highly imbalanced (1 (2225 data points), 2 (200), 3 (1158), 4 (1467), 5 (5716), 6 (1026), 7 (4101), 8 (6178), 9 (5442), 10 (5200), 11 (1614) and 12 (4404)). For each time point, we performed a witness sampling of 200 data points since it was the lowest number of samples among all time points. We identified the simplicial complexity to vary over the time, suggesting a potential better summary statistic with better distinction among the time steps (Figure 7). The normalized simplicial complexity (computed as the ratio of the number of simplicial complexes discovered within the data to the number of those discovered within the null model) suggested an abundance of high-dimensional simplices over the null models. The existence of a significant number of high-dimensional simplices was observed for the first time at the single-cell level. In all time points, the  
Figure 5: Dimension reduction of the dataset. PCA, TSNE, and UMAP applied to our preprocessed gene expression profiles. number of simplices of dimensions larger than one in the null model was far smaller than those found in the actual data. In addition, we observed relative differences between what we discovered in the null models, and the actual data increased drastically when the dimensions were higher. Furthermore, the number of low-dimensional simplices (up to three dimensions) of the data appeared to be equal to or smaller than that of the null models (with a normalized complexity less than one), suggesting a possible transfer from a lower-order clique structure to a higher-order structure. In order to investigate the trade-off between the higher-order and lower-order simplicial complexity in the developmental stages, we mapped the normalized 3-simplicial complexity against the normalized 1-simplicial complexity.