Independence Complex of Kneser Graph

An independence complex of a graph is the simplicial complex consisting of the independent sets of the graph. We’ll discuss the topological properties of the independence complexes of Kneser Graph $KG(n, k)$ using  Vietoris-Rips complexes. Barmak gives the homotopy types of the independence complex of $KG(2,k)$ which is a wedge sum of spheres. For $n\geq 3$, not much is known about the independence complex of $KG(n, k)$.   We’ll investigate the maximal simplices in such complexes and then provide a lower bound on some specific dimensional homology.