KNOT IDENTIFICATION (geometry / topology)

Knotting Concepts & Explorations
1

To my audience of one - Dan!

So as not to de-rail another topic thread, I've started this new topic.

Please examine all of the knots in the image below.

I am introducing the idea of taxonomic knot classification.

This is conceptual stage only, very much a work in progress!

Domain: Knots

Class: Fixed eye; Joining; Hitch

Order: End of line; Mid-line; Movable/Adjustable

Family: Symmetry (Symmetric, Asymmetric, Structure, Shape and Form)

Genus: Load response (behaviour under various loading profiles)

Species: Individually identifiable knot (eg Zeppelin bend)

Mapping example:

Domain: A Zeppelin bend is in the domain of 'knots' - its a knot.

Class: It is a joining knot (end-to-end joint)

Order: End of line - joining knots are formed in the ends of ropes/cordage (ends are united)

Family: Symmetric

Genus: Non jamming

Species: Zeppelin bend

.................................................................

In the image below, figures 1,2,3 show response to load. All are Zeppelin bends at varying stages of compression and deformation, yet they are still identifiable as Zeppelin bends. In other words, figure '1' is the same species as figure '3'.

In the case of figure 'A' and 'i' - we can make the claim that they both are from the same class, order, family, and genus. But, are they the same 'species'?

'A' is the same species as '1'.

However, 'A' is not the same species as 'G'.

Happy to discuss and thresh out the semantics and concepts...

In the lower image:

'A' is the same species as 'B'. A chiral apposite (mirrored form) is the same species. If chirality was an issue, all knot book authors would show 2 versions of every knot - thereby virtually doubling the volume of their publication. Ashley appears to only show right-handed chirality 'Bowlines' (with few exceptions) - although I will need to thoroughly check this.

Knot Classification_Zeppelin
Knot Classification_Zeppelin1658×2189 466 KB

Zeppelin-Mirror opposite
Zeppelin-Mirror opposite1090×1220 146 KB

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2

Not an audience of One. I have nothing to add Mark.

3

Good to know Scott!

By the way - your 'Scotts locked Bowline' has been gaining a lot of ground around the world. It is becoming the No. 1 'go to' knot for life critical applications.

It is simple and effective - and fit-for-purpose.

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4

Glad that it is being used. Such a simple lock.

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5

Interesting idea, and I think you're right on two key points: a knot stays the same species however hard it's loaded or deformed (your figs 1–3), and a mirror-image form is the same species. Otherwise every knot would come in two.

A few thoughts on the upper levels:

It isn't really a hierarchy. In biology each rank nests inside the one above. Here the levels are independent attributes. A knot can be both adjustable and tied in the bight, yet "Mid-line" and "Movable/Adjustable" are separate, mutually exclusive Orders. It might work better as a set of independent facets (type, how it's tied, symmetry, load behaviour) than as a tree.

Genus conflicts with your own example. If "non-jamming" sits above species, a Zeppelin that jams in some loading case would have to change genus, while you say deformation doesn't change species. Load response looks like a property of a species rather than a rank above it.

Family mixes several things. Symmetry, structure, shape and form are different axes.

The classes need widening. Binders, stoppers, lashings, shortenings and decorative knots don't fit yet, and the line between eyes and hitches needs drawing (e.g. a Clove Hitch tied in the bight).

Species needs a test. You say A is the same species as 1 but not as G. What decides it? I'd suggest topology, meaning how the parts interlace, independent of dressing and setting. That criterion is really the heart of the whole system.

Happy to keep threshing it out.

6

Thanks for re-activating this topic.

As per my other post - we have to be careful because there are some people who view 'arguments' as being 'bad'. That is, if you have an opposing argument, you are 'bad'. I see opposing arguments as being 'good'. That's why I like engaging with Dan because it tests the underlying logic and ideology.

The English language is complex - over the past 20 years or so, Dan has shown me that I need to be very clear in how language is applied.

With regard to 'family' - I threw it out there for want of a better term... the idea being to consider the structure of a knot and how to classify it. The taxonomic order does not map one-to-one onto my idea - think of each taxonomic descriptor as a 'placeholder'.

Some knots are symmetrical while others are asymmetrical. For example, the Butterfly is asymmetric while the Zeppelin bend is symmetric. And yet both are formed from 2 interlinking simple overhand knots.

In terms of eye knots - the F8 (Ashley #1047) is symmetric, while a simple Bowline (Ashley #1010) is asymmetric. Both are 'eye knots'.

Fundamentally, all knots can be grouped into the following 'categories':

Eye knots

End-to-end joints ('bends')

Hitches (require a 'host')

I am excluding decorative knots.

And I need to consider mid-line knots versus end of line knots... and how to tie this all together into a coherent, logical order.

Load response is something I was considering... eg a Crossing hitch (aka Munter hitch) capsizes to align with the load path. A Carrick bend also capsizes when load is applied - and reforms into two interlinking Munter hitches (Crossing hitches).

Some knots are resistant to jamming - others jam.

I don't think 'topology' (mathematical expression of knots) is the way forward. The number of crossing points tells you nothing about how a knot performs as a tool. Rather, geometry (geometric form) is important.

I used the taxonomic method as a basis for forming the underlying theory. It may be the wrong underlying idea... but I think its worth exploring further to see where it leads...

7

Thanks, and I agree completely that opposing arguments are good. It's how ideas get tested. I'm here to learn, not to win.

Treating the ranks as placeholders makes sense, and I think your three basic categories (eye knots, end-to-end joints, hitches) are a sound starting point.

On topology, I think I explained myself badly. I agree that crossing numbers tell us nothing about how a knot performs as a tool. What I meant was closer to what you call geometry: which parts go where, and how they interlace. Your Butterfly/Zeppelin example shows it well. Both are made from two overhand knots, yet they are clearly different knots because the parts are arranged differently.

So maybe we're nearer each other than it looked. The question I find most interesting is how you would put that into words: what exactly makes A a different species from G, while 1 and 3 are the same despite the deformation? If we can pin that down, I think the rest of the system will fall into place.

Looking forward to seeing where it leads.

8

Knots are a hard thing to come to grips with,
to organize into some understandable structure.

My efforts of late have been working with the idea
that a knot is a tangle with a Load_Profile.
With this inchoate conception in mind, the Sheet-bend
/Bwl/usual_net-knot group seemed to work fairly well.
But I came to regard this somewhat happy result as
specious : that these knots worked well in that they
fit a pretty planar depiction from which to see their
Tangle & make the workings --but even so, there were
problems (e.g., in loading Bwl Tail vice S.Part, how is
that now-opposing-eye_legs Tail to be brought out
from "between" the eye legs?!).

With a 2-tangle --and I've pretty much looked at
only knots having this--, there are 8 EKs, 4 e2e Joints,
4 End_1 -v- End_A + B (think "sheet bend to eye"),
4 End_1 -v- 3_other-ends, and I guess 4 stoppers
... and more. And this doesn't quite allow for what
I borrow as "transvers loading" (for me : a wide-spread
EK) of EKs.

Talk about not changing "species" :: even if there's
a distortion/capsize of e.g. a Bwl EK into a Pile-Hitch
Noose (i.e., the S.Part-to-OELeg straightens (or nearly)
and REL & Tail now make all/most of the *knotting* !?

With the 1-tangle of an Anchor Bend, if this it
Tail-loaded it's pretty much like a nipping loop
around Port.Bwl 2 Eye Legs --Ashley at some
point more or less acknowleges this but saying
it's often tied too tightly (and thereby acting more
hitch the twin EK).

SOOO much is so variably indicated; so many materials
and dressings & settings, loads, and ... .

(--just closing thought, to a practical concern :
for abseiling, if the e2e Joint comes to arrest by
some rock bit on pulling down an abseil rope,
increased pulling on the one rope will more
have practical STOPPER effect with rising
force, as that force isn't shared by the joined
upper rope, just resting pretty much, as the
knot is pulled by lower rope.
.:. an odd sort of stopper, in having 2 strands
(pulled 1-2 vs. resting A-B), but it so functions thusly.

--dl*
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