The Bowline Knot is corrected by completion

[Stagehand]

The Bowline Knot is corrected by completion.  The Bowline Knot does not form a complete circuit.  If it is to be identified in the Mathematical Knot Tables, a complete circuit must be inferred.  Seeing the Bowline as a Partial Carrick knot that may be finished like a Carrick Mat completes the Bowline as Rolfsen 8_20.
It is interesting that Rolfsen 8_20 begins a most rapidly increasing series, the series of non-alternating prime knots.  By itself, Rolfsen 8_20 serves its purpose as an invariant.
Yet for me the remarkable feature available for the Bowline Knot as the complete circuit is that now the Bowline Knot may have a generalized path toward sufficiency through completion toward doubling.  It would be the remark of an expert who will say that the correct way to continue the Bowline Knot is with the pictured knot, the Safety Bowline.

[Dan_Lehman]

The Bowline Knot does not form a complete circuit.  If it is to be identified in the Mathematical Knot Tables, a complete circuit must be inferred.  Seeing the Bowline as a Partial Carrick knot that may be finished like a Carrick Mat completes the Bowline as Rolfsen 8_20.

Most cases of completing the circuit in this way
(of joining tail to SPart) for the bowline found
in the wild would give a mathematical knot
of six crossings --viz., Rolfsen 6_3 (not 8_x)--,
because of how the draw of the SPart positions
the tail (which is how it flows into the Yosemite bwl.).
I.e., the tail lies between the eye legs, and so it must
be drawn either over or under a leg to close the circuit,
and it makes a difference which way it goes!
(And this was my point to asking by what rule you
had made the conversion.  Xarax, note, would choose
paired links for eye knots & end-2-end knots:
for this, the result is quite surprising --minimal!!
--or is it?  >> The squaREef knot dissolves out of
interlocked links (think of "girth-hitching" slings).)

Even at *only* 6 crossings --sitting nearly at the
top of the table (!)--, it can be quite a brain pain
to determine this match (indeed, initially I had
it as seven --then innocently rearranging ...,
I found that there were only six).


--dl*
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