Homotopy types of Vietoris-Rips complexes of Hypercube Graphs

We describe the homotopy types of Vietoris-Rips complexes of hypercube graphs at scale $3$. We represent the vertices in the hypercube graph $Q_m$ as the collection of all subsets of $[m]=\{1,2,\dots ,m\}$ and equip $Q_m$ with the metric using symmetric difference distance. It is proved in \cite{AA22} that the Vietoris-Rips complexes of hypercube graphs $Q_m$ at scale $2$, $\mathcal{V}\mathcal{R}(Q_m;2)$, is homotopy equivalent to $c_m$-many spheres with dimension $3$ where $c_m={\sum }_{0\le j<i<m}(j+1)(2^{m-2}-2^{i-1})$. Questions are raised in \cite{AA22} for determining the homotopy types of $\mathcal{V}\mathcal{R}(Q_m,r)$ with large scales $r=3,4,\dots ,m-2$. We prove that for $m\ge 5$, $$\mathcal{V}\mathcal{R}(Q_m;3)\simeq (\underset{2^{m-4}\cdot \left(\genfrac{}{}{0px}{}{m}{4}\right)}{\bigvee }{S}^7)\vee (\underset{\sum _{i=4}^{m-1}{2}^{i-4}\cdot \left(\genfrac{}{}{0px}{}{i}{4}\right)}{\bigvee }{S}^4)\mathrm{.}$$