Fig. 1

Example for topological structures of simplicial complexs. Consider 2-simplex \(S = \{ v_0, v_1, v_2\}\), and its boundary-faces \(S_0 = \{ v_1, v_2 \}\), \(S_1 = \{ v_0, v_2 \}\), \(S_2 = \{ v_0, v_1 \}\). Suppose that simplicial complexes at time-step 0 and time-step 1 are \({\mathcal K}_0 = \{ S_0, S_1, S_2, \{ v_0 \}, \{ v_1 \}, \{ v_2 \} \}\) and \({\mathcal K}_1 = \{ S, S_0, S_1, S_2, \{ v_0 \}, \{ v_1 \}, \{ v_2 \} \}\), respectively. Then, the first homology groups are given by \(H_1 ({\mathcal K}_0) = {\mathbb Z}\) and \(H_1 ({\mathcal K}_1) = \{ 0 \}\). Namely, \([{\mathcal K}_0]\) has one 1-dimensional hole, while \([{\mathcal K}_1]\) has no 1-dimensional holes. This implies that the topological structures of \([{\mathcal K}_0]\) and \([{\mathcal K}_1]\) are different, and \({\mathcal K}_0\) and \({\mathcal K}_1\) are qualitatively distinct