  This paper aims to determine the images of the braid group under
representations afforded by the Yang Baxter equation when the solution is a
nontrivial $4 \times 4$ matrix. Making the assumption that all the eigenvalues
of the Yang Baxter solution are roots of unity, leads to the conclusion that
all the images are finite.
  Using results of Turaev, we have also identified cases in which one would get
a link invariant. Finally, by observing the group algebra generated by the
image of the braid group sometimes factor through known algebras, in certain
instances we can identify the invariant as particular specializations of a
known invariant.
