The Bowline is defined

[Stagehand]

The Bowline is defined as an end reeved through a turn, then continued through a three-line crossing, and then directed toward an evident circuit. 
Once an end is reeved through a turn, any resulting tying diagram that returns through the turn, will not have complete over-under patterning and will instead have ambiguous patterning.  Likewise, any tying diagram with a three-line crossing will have ambiguous patterning through this crossing.
These ambiguous tying patterns are suitable for representing the Bowline Knot and its variations.  These patterns generate satisfactory and unsatisfactory knots.  An interesting case is that of a knot that has the same outline as the Bowline Knot, ABoK #1010, yet it has a different over-under patterning.
As a class, these defined bowlines are each and all made better by continuing the ends into circuits.

[Dan_Lehman]

any tying diagram with a three-line crossing will have ambiguous patterning through this crossing.

This is not true : one can represent the knotted material
with parallel lines, and this enables a multiple-line crossing
to be disambiguated (within reason of the "multiple" --which
might be as few as 4 or 5, but done readily enough with 3).
In any case, one can aid comprehension by labeling the segments
and specifying their relative "over/under" relations (e.g., "1-2"
goes UNDER "6-7") --as well as indicating connectivity!

That all said, one might do well to orient knot parts so as to
avoid this problem --unless doing so too distorts a natural view.


--dl*
====

[xarax]

   I like the idea to start from a few simple, general, yet still ambiguous patterns, and, by removing the ambiguities, get all the possible unambiguous forms there exist. In particular, by specifying the over/under relations at this(those) three-line crossing(s), you attempt to arrive at a set of completely specified, so unambiguous diagrams, representing all the possible knots which belong to this category.
   It seems to run towards the opposite direction of which one might had expected... In our attempt to "save the phenomena" of the complex natural word around us, we are used to start from a few completely specified simple things ( which we call "axioms" or "elements" ), and then, by combining them in some proper way, we try to remove the ambiguity of the structure of each complex thing we want to "explain", be it a theorem or an object. Therefore, the removal of ambiguity is achieved at the end of the process, it is our task, not our method, as in the "knot generator" you describe here. Innovative, and very interesting ! 
  ( However, I am not sure that all bowline-resembling end-of-line PET loops can be produced by this "bowline generator". Alan Lee, for example, has tied and presented in this Forum dozens of dozens of new such loops - I do not see how they can be included in your scheme, because the Working Part follows some new and unpredicted ( by me, at least ) paths ! )

[Dan_Lehman]

   I like the idea to start from a few simple, general, yet still ambiguous patterns, and, by removing the ambiguities, get all the possible unambiguous forms there exist.

Starting with an ambiguous representation can lead
to new discoveries!     Seriously, sometimes it is
helpful to look as a drawing and mentally make new
connections (irrespective of whether the drawing is ambiguous).
(The ensuing problem is that then one might have even more
"new knots" to record --the pile grows, rather than be reduced,
of one's knot-recording workload!)

--dl*
====