  A Lorenz link is equivalent to a T-link, which is a positive braid built by
concatenating torus braids of increasing size. When each torus braid except the
largest is obtained by full twists, then the T-link can be described as the
Dehn filling of a parent link. In this paper, we completely classify when such
parent links are hyperbolic. This gives a classification of the geometry of
T-links obtained by full twists when the amount of twisting is large, although
the bound on the number of required twists is not effective. We also present
effective results on hyperbolicity for two families of T-links obtained by
twisting, even when the number of twists is small. Finally, we identify
families of satellite T-links obtained by half-twists.
