Vietoris-Rips complexes of Platonic solids (Q6581285)

The paper determines the homotopy type of the Vietoris-Rips complexes of the vertex sets of the Platonic solids \(VR_r(P^{(0)})\) (Main theorem). A combinatorial distance of VR complexes is given in Table 1. Although most cases are not hard to compute (Lemma 2.1; Lemma 2.4) as the authors noted, several statements with regard to the dodecahedron are (\(\delta_2 < r \leq \delta_3\) and \(\delta_3 < r \leq \delta_4\)). The result of the former case is given in Lemma 2.6, and the latter one is given in Proposition 4.4. The main idea of the computation is to\Napply discrete Morse theory to investigate the change of homotopy type.\N\NThe main tool of discrete Morse theory used is as follows: (Theorem 1.1) Let \(K\succ K'\) be simplicial complexes and \(K \setminus K'\) be finite. Then, \(K \searrow K'\) if and only if there exists a complete acyclic matching on the set of all simplices of \(K\) that are not contained in \(K'\).\N\N\NNotable statements include:\N\begin{itemize}\N\item Let \(F\subset \mathbb{R}\) be a regular polygon, \(T\) be a triangulation of \(F\). Let \(\Delta\) be the simplex on the vertex set of \(F\), then there is a deformation retraction from \(\Delta\) to \(T\). (Lemma 2.6).\N\NIn the proof of this statement, the authors construct an acyclic matching on the simplices of \(\Delta\) that are not in \(T\) and apply Theorem 1.1 to show \(\Delta \searrow T\).\N\N\item Let \(P\) be the dodecahedron. There is a strong deformation retraction from \(VR_{\leq 3}(P^{(0)})\setminus \{ \tau_1 \ldots , \tau_{10}\}\) to \(VR_{\leq 2}(P^{(0)})\). (Proposition 4.2).\N\end{itemize}