We live in a New York neighborhood that doesn’t seem like a New York neighborhood. Battery Park City is a relatively new notch added on the lower west side of Manhattan, built in the 60s and 70s from earth excavated for the World Trade Center buildings and a few other large-scale public works. Head west across the West Side Highway, and you leave TriBeCa for blocks that feel like a concoction of city and suburb. A good third of the area is given over to…