  The ``Links-Gould invariant'' is a two-variable Laurent polynomial invariant
of oriented (1,1) tangles, which is derived from the representation of the
braid generator associated with the one-parameter family of four dimensional
representations with highest weights (0,0|a) of the quantum superalgebra
U_q[gl(2|1)]. We use an abstract tensor state model to evaluate the invariant,
as per the construction of the bracket polynomial state model used by Louis
Kauffman to derive the Jones polynomial. This model facilitates both
computation and theoretical exploration.
  Our family of representations has a two-variable quantum R matrix (unique up
to orthogonal transformations). Choosing this R matrix to yield the
representation of the braid generator ensures that our polynomial invariant
will also have two variables. We construct this R matrix from first principles.
We have evaluated the invariant for several critical link examples and numerous
other links of special forms. Throughout, the assistance of Mathematica has
been invoked.
  We observe that the Links-Gould invariant is distinct from the two-variable
HOMFLY polynomial in that it detects the chirality of some links where the
HOMFLY fails. Notably, it does not distinguish inverses, which is not
surprising as we are able to demonstrate that no invariant of this type should
be able to distinguish between inverses. It also does not distinguish between
mutants.
