  In this paper, we study the basic properties of persistent homotopy groups.
We show that the persistent fundamental group benefits from the Van Kampen
theorem, and the interleaving distance between total spaces is less than or
equal to the maximum of the interleaving distances between subspaces. We also
prove excision and Hurewicz theorems for persistent homotopy groups. As an
application, we analyze the sublevelset persistent homotopy groups of the
energy landscape of alkane molecules.
