Uncountably many inequivalent Lipschitz homogeneous Cantor sets in ℝ3

Abstract

General techniques are developed for constructing Lipschitz homogeneous wild Cantor sets in ℝ3. These techniques, along with Kauffman's version of the Jones polynomial and previous results on Antoine Cantor sets, are used to construct uncountably many topologically inequivalent such wild Cantor sets in ℝ3. This use of three-dimensional finite link invariants to detect distinctness among wild Cantor sets is unexpected. These Cantor sets have the same Antoine graphs and are Lipschitz homogeneous. As a corollary, there are uncountably many topologically inequivalent Cantor sets with the same Antoine graph.