It is true that the TIB knots are fewer than the rest, but that does not mean that they are few ! 
Here, to be accurate, one needs to qualify with "interesting"
or something, of "knots" --for by some simple (and quite
UNinteresting) methods, it can be shown that there are
some infinity of knots ... : equal in (impractical) number!
(E.g., re-tucking an eye through the knot body begets yet
another "knot" --unlikely to be anything but a theoretical
proof.)
A possible simple "base" that we can exploit to tie midline double loops,
is the two nipping loops and two bights formation, I have called "eyes and ears"
, reported at (1).
1) http://igkt.net/sm/index.php?topic=3020.msg21737#msg21737
This is significant, and IMO significantly urges that such
discussions be in the
Knotting Explorations forum and
not
Practical --and we know the old arguments about
this, and
Why tie ...? vs.
Because someday ... ping pong.
This is a good step of pulling from knots such a "basis"
for knot construction, and employing it to take one
further into the knots universe. Yes, there are some
interesting ways to engage rope in its middle, with
some general bases such as Xarax presents. One
can, e.g., elaborate on Ashley #1408 & 1425 (and
more) schemas, out of the *middle*, for interesting
"twin-eye" knots --as I envision them (where the two
eyes are roughly adjacent and used qua one, so to get
assured behavior when loaded in either direction (as
contrast w/e.g. the
butterfly's asymmetry)).
!! There are some very interesting directions to take
by such elaboration and then exploration of *bases* !!
