On Defining Knots

The blogs in this series are being offered to stimulate reason… not by preaching or propaganda, but by example, by following reasoning in action  —as it “works its magic”, as it “makes a difference” or as it “shines a light”…  onto otherwise unexpected, unpromising mathematic dark corners. It is a truism that reason could be applied, in principle, to any new, complicated, fully defined, situation. But we are living in a time of unprecedented technological change, when complex projects are burgeoning everywhere. Reason can bring great illumination, shedding much needed light onto mathematics (and via mathematics onto the world) but…   it isn’t doing as much as it could, because it has sadly become a neglected art. It sometimes seems as if the impulse to reason in mathematics is fading away. (And sans cognition, mathematic process reduces to bare ‘handle turning’, hardly more taxing —probably— than driving trains, or putting airliners onto automatic.). In the past reasoning has been the secret of maths’ success, but it won’t maintain this record if we let it fade.  This instalment (28) is about the application of reason to knots.

Mathematic problems tend to arise from the flummoxing effect of complexity… (This is complexity in well-defined i.e. deterministic, situations.) Their definition tells us that they ought to be easy to read. But the fog they provoke tells us that they’re not.  It is quite odd, actually, and a rarely mentioned fact, that the commonest of such problems are knots.  Perhaps knots should be the first item on any mathematic curriculum.  Every home, probably, has a cupboard, drawer or old carboard box containing a daunting tangle of string, twine or flex in the form of a knot.  It poses the unspoken challenge: “How to sort out this ungodly tangle?”.

I attended a lecture long ago (1966) at Harrow School with a small party of volunteers (sixth-formers) from the school where I was teaching at the time.  The speaker was Prof. Christopher Zeeman. His exposition was a simplified introduction to the theory of knots. A large, enthusiastic audience had assembled from schools across North London. He started with simple examples, such as granny knots, reef knots and simple fisherman knots.  It was my first introduction to knot theory. During an hour’s lecture, he asked some simple questions… showed rules, moves, and isomorphisms with knots. It was a robust introduction to the topic, and I should think that most of the students present were stimulated by it.

But…was something missing? Afterwards, on reflection, I felt that there was: Zeeman’s start was surely much too ad hoc.  Some examples of rudimentary knots were shown, but where was the generality? Where did these particular knots fit in? Were they typical? Did all knots fall under these specific rules and moves? Later, I began to ruminate on what we had been shown. I was quizzing myself on how one could arrive at a definition of <<A general knot>>. A weekend of intense cogitation followed. On the Monday I wrote to Zeeman with a short, hasty note, containing the gist of the account set out in this blog.

(Zeeman didn’t respond, he was probably very busy. I was also busy. Priorities I considered more urgent were dominating my attention —both then and for many years afterwards. As a result, it has taken so long for my thinking arising from that distant evening at Harrow to renew, gel, and eventually to end-up here.)

Reasoning led me to think that we ought to be able to find a canonic method for constructingknots. What were the building blocks? What were the “weaving actions” which turned a straight rope into a knot?

My first thought was that a piece of string of indeterminate length and involved in numerous eventual contortions ought to be anchored. So I started off with a piece of string like this:

O |—————————————————————————à F

[DIAGRAM 1]

The LH end (O) was the anchor point, and the arrow was the “free end” of the rope, F.  The rope is treated as if it is lying on a horizontal plane.

Zeeman had shown us some closed knots where the string formed a continuous loop.  But these, I reasoned, could be addressed at a later stage, after first dealing with anchored knots. They could be accommodated fairly easily, by identifying O and F. Next I conceptualised a discrete, distinctive, local ‘knot-step’ or ‘wrap’, during which F wrapped itself around a particular bit of the open rope (T₁), starting above T₁. It would either be clockwise of n turns or anticlockwise of –n turns.  (T₁ is the first ‘wrap’.  ‘Clockwise’ is defined as being as seen when one is looking along the direction of the rope treated as a path from O to F. All turns (‘wraps’) start from a point above the plane on which the rope is resting.  A turn of -1 is, in effect, a passage of F under the rope at this point.)

So now we had a situation like this:

[DIAGRAM 2]

It could be initially represented by the symbolic device: O x T₁(x) x F, where each ‘x’ stood for a length of open rope. The rope was now divided into three sections (a) from O to T₁, (b) round the loop starting at T₁, (c) between T1 and F.

The second wrap, T₂, could be woven (constructed) in any one of these three available lengths of open rope.  For example, T₂ could be on the final third of the rope, and a knot involving this would be denoted by the device:

O x T₁(x) x T₂(x) x F.

There are now five available sections of open rope on which the next wrap, T₃, could be turned.  Two of these sections of string may be described as ‘loops’ as shown in Diagram 3.

[DIAGRAM 3]

Now the third wrap, T₃, could be allocated to five different sections, two of which are loops. So, developing the existing example,  in the case when the third wrap is on the original loop we have a nesting of brackets, coded like this:

O x T₁(x T₃(x) x) x T₂(x) x F, and as shown in Diagram 4.

[DIAGRAM 4]

IMPROVING THE NOTATION

There are two further data items to be added: first the number of clockwise turns involved in each wrap (n) shown as a superscript, say T₁² (for two clockwise turns),  and second the order of the rope sections, starting from x₁ at O.

If the number of turns round the basic rope doubles each time, starting with 2 turns, and reverses sign each time, the full code for the knot shown in Diagram 4 is this:

O x₁ T₁²(x₂ T₃⁸(x₇) x₃) x₄ T₂^-4(x₅) x₆ F.

The simplest possible knots are incidentally denoted in this notation:

O x T₁^-2 x F    and O x T₁²  x F      [See DIAGRAM 5]

[Such a symbolic device can eventually serve as an unusual ‘algebraic object’ capable of being subject to operations such as (i) adding one knot to another (the F of the first knot becoming the O of the second), (ii) subtracting, multiplying, dividing, etc.  such knots (iii) simplifying the coding of a specific knot to reflect self-evident consolidations of the wraps, (iv) determining unwrapping procedures for a given knot.  See the final section of this blog.]

It will be noticed at once that the number of “open sections” of rope increases by 2 after each new wrap. The knot shown in Diagram 4 has 7 open sections of rope… onto which a fourth wrap, T₄ could be placed.  A four wrap knot will have nine open sections of rope.

And so on…

The schema proceeds in this fashion until the last wrap T_L is set. It should be noted that, at a typical moment during the process,  after setting a new wrap, the leading part of the rope can be conceptualised as being able to move freely above the current ‘incomplete knot’ to set up a further wrap on any section of the free rope available.  This is a topological configuration and there is a lot of flexibility built-into the schema.

A KEY QUESTION

Does this construction schema cover all possible knots? Yes, it is self-evident that the first “wrapping action” must occur somewhere on the rope, thus creating three open sections of rope on which future wraps might be set. The second “wrapping action” will likewise occur on one of these open sections of rope, thus creating five new open sections of rope for future wraps…. and so on. These successive choices cover all the options.

After completing the construction schema, the possibility arises of some consolidation of adjacent loops (see below).

It might be wondered whether a new wrap could be specifically created —contrary to previous practice—  by weaving round an existing wrap, thus compromising the closure of the schema as claimed above. But the flexibility mentioned allows for this to be explicitly resisted. Instead of trying to weave a new wrap around an existing wrap, the schema can build-in the requirement that such a move is made onto and over the two sections of open rope to the Left of the ‘existing wrap’. See Diagrams 6a, 6b.

[DIAGRAMs 6a, 6b]

In Diagram 6a the free end F has attempted to wrap round an existing wrap, T.  This will be declared inadmissible in the construction protocol. Instead the free end of the rope is displaced to the Left of T as in 6b, and finally shown (in this particular example) as four minimal wraps in 6c.

DEVELOPMENT

It appears that a new kind of algebra will be needed to study the equivalence of two different knot codes A and B when the knot represented by A is topologically equivalent to that denoted by B. An axiom set of such equivalences will be needed to reduce the final schema to an unique, reduced, canonical form, which could serve as a definitive classification for knots.  This ‘knot algebra’ will be discussed in the October Blog.

The simplest consolidation arises when a finalised knot contains two successive wraps T_aand T_b , T_a creating a new loop and the second T_b being on this new loop (See Diagram 7). Provided that the final knot contains no further wrap between T_a and T_b , the two wraps can then be combined into a single wrap, which will of course have the sum of their respective turns.  In a case when the two wraps T_a and T_b have the same number of turns, one clockwise the other anti-clockwise, the two wraps disappear!

[DIAGRAM 7]

A broad sign of the structure of such a classification is that every knot will be representable by a non-reducible trifurcating root tree of the kind illustrated in Diagram 8.

[DIAGRAM 8]

The seven free branch ends of this trifurcating tree x₁, x₂, x₃, x₄, x₅, x₆, x₇  are places from which new trifurcations can grow.

CHRISTOPHER ORMELL 1^st September 2022