Crossing number of (closed) homogeneous braids

The Polyak and Brandenbursky invariants are applied to estimate the crossing number of (closed) braids and extend the previously known minimality criteria for diagrams of positive and alternating braids to homogeneous ones. In particular, it is proved 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 developed here is conceptually similar to recognizing alternating links on the basis of the Tait conjectures.