  We investigate the Gram determinant of the bilinear form based on curves in a
planar surface, with a focus on the disk with two holes. We prove that the
determinant based on $n-1$ curves divides the determinant based on $n$ curves.
Motivated by the work on Gram determinants based on curves in a disk and curves
in an annulus (Temperley-Lieb algebra of type $A$ and $B$, respectively), we
calculate several examples of the Gram determinant based on curves in a disk
with two holes and advance conjectures on the complete factorization of Gram
determinants.
