I will not try to explain to you that, to define the relation 1+1=2, Principia Mathematica needed/used 150 pages of difficult Set Theory definitions..
And was seriously *dented* by far fewer, but resounding pages
from Goedel, where Hilbert & Frege also tried hard but failed.
Yes, defining the "simple" is ... not so simple!!
(And these were bright thinkers amongst bright thinkers!)
Can you see the shaping of "knot" definitions et cetera as activities
in this realm of deliberation, distinguishable from just putting up
a knot-form for consideration of (possible) practical use?
In fact, I might suggest that it is as well a fit to a "theory" (and
maybe a better name), philosophical --but *beside* "practical"--
heading that one simply explores the *knot space* of structures
that can be roughly conceived as points in some vast matrix,
a knot universe (multi-dinensioned). Some things can be stated
and projected as series: the
overhand ("pretzel"), and next
a dble.overhand and so on, but then in the multiple forms one
has different orientations (anchor bend, stangle knot, ...?) of the
same topology.
Considerations of which seemingly "simple" things can quickly
be frustrating (to me, at least), as interconnections abound.
Of course, the mere
mention of some aspect is not throwing
things off; that is different from where the thread topic focuses
on that aspect, in contradistinction to something "practical".
--dl*
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