  We introduce a matrix representation of a chord on a tangle which leads us to
representing tangle chord diagrams as stacks of matrices that we call books. We
show that band sum moves, Reidemeister moves as well as orientation changes are
implemented on \widetilde{Z}_f - a framed link invariant constructed from the
Kontsevich integral that's well-behaved under band sum moves - by matrix
congruences. We prove that being given the bare framed Kontsevich integral
Z_f(L) in book notation for some unknown link L, we can determine what the link
L is, as well as the projection of Z_f(L) in the original completed algebra of
chord diagrams.
