The topological properties of Julia sets play an important role in the study of the dynamics of complex polynomials. For example, if the Julia set J is locally connected, then it can be given a nice combinatorial interpretation via relating points of J and angles at infinity [DH84]. Moreover, even in the case when J is connected but not locally connected, it often admits a nice locally connected model - the so-called topological Julia set with an induced map on it - which is always locally connected, similar to polynomial locally connected Julia sets, and has the same combinatorial interpretation as they do (Kiwi [Ki04] proved this for all polynomials with connected Julia sets but without Cremer or Siegel periodic points). This connection makes the study of both locally connected polynomial Julia sets and topological Julia sets important.