Crossing number of (closed) homogeneous braids

We apply the Polyak and Brandenbursky invariants to estimate the crossing number of (closed) braids and extend the previous minimality criteria for diagrams of positive and alternating braids to the case of homogeneous braids. In particular, we prove that a diagram of a homogeneous braid is minimal if and only if this diagram is homogeneous. These results lay the groundwork for a potential solution to the recognition problem for homogeneous knots and links. The approach we develop is conceptually similar to the method of recognizing alternating links based on Tate conjectures.