  The Gram determinant of type $A$ was introduced by Lickorish in his work on
invariants of 3 - manifolds. We generalize the theory of the Gram determinant
of type $A$ by evaluating, in the annulus, a bilinear form of non-intersecting
connections in the disc. The main result provides a closed formula for this
Gram determinant. We conclude the paper by discussing Chen's conjecture about
the Gram determinant of type $Mb$ evaluated in the Klein bottle.
