Milnor invariants and thickness of spherical links

Various authors have studied the question of how long a rope of a given thickness is needed to tie a given isotopy class of knot or link. In joint work with Rafal Komendarczyk and Fedya Manin, we generalize this work to spherical links in arbitrary dimensions. In more detail, we study their Milnor invariants in terms of Massey products and prove asymptotically optimal upper bounds on Milnor invariants in terms of embedding thickness. We find a dichotomy between polynomial and exponential bounds, depending on the dimensions of the spheres. We apply our results to answer a question of Freedman and Krushkal about exponentially thin 2-complexes in 4-space.