Can you turn Nalebinding into a Turing machine?

Is Nalebinding (suitable to use as) another ancient Turing machine? In a previous series of posts I argued that the three Moirai (aka the Fates) have this capability, see:

From wikipedia page about Nalebinding:

“Nålebinding (Danish: literally “binding with a needle” or “needle-binding”, also naalbinding, nålbinding or naalebinding) is a fabric creation technique predating both knitting and crochet. Also known in English as “knotless netting,” “knotless knitting,” [1] or “single needle knitting,” the technique is distinct from crochet in that it involves passing the full length of the working thread through each loop, unlike crochet where the work is formed only of loops, never involving the free end.”

“Nålebinding works well with short pieces of yarn; based on this, scholars believe that the technique may be ancient, as long continuous lengths of yarn are not necessary. The term “nålebinding” was introduced in the 1970s.[1]

The oldest known samples of single-needle knitting include the color-patterned sandal socks of the Coptic Christians of Egypt (4th century CE), and hats and shawls from the Paracas and Nazca cultures in Peru, dated between 300 BCE and 300 CE.[2][3]

Here is a pair of ancient egyptian socks (courtesy this wiki page) made by the nalebinding technique.

socks

OK, so this is an ancient technique which bears a recent Danish name. I googled for math related to nalebinding and I got some links pointing to hyperbolic surfaces and other nice math visualisations (try to  google for your pleasure), but actually I am afraid of giving the said links because nalebinding has a huge fan base which might get attracted to this post by inadvertence. This post is not, I repeat NOT about the nalebinding hobby.

According to what I could grasp from this very interesting page, there exists a scientific notation for nalebinding stitches which was introduced in the article (cite from the said page):

Hansen, Egon H. “Nalebinding: definition and description.” Textiles in Northern Archaeology: NESAT III Textile Symposium in York 6-9 May 1987, ed. Penelope Walton and John P. Wild, pp. 21-27. London: Archetype Publications, 1990.

Presents a notational system for describing nålebinding stitches that is based on the course the thread takes in traversing each stitch. No historical information included, but lots of technical plates of interlacement variants.

[I need a copy of this article. I could not reach it until now, please, could anybody send me a pdf?]

I think Nalebinding could be turned into a Turing machine. Here comes the more technical part.  Indeed, in graphic lambda calculus the main move is the graphic beta move (which corresponds to beta reduction in lambda calculus).

betar

Or, this graphic beta move can be put into the form of a braiding move. First we define the crossing macro:

betar_dif_11

then we re-write the graphic beta move as:

betar_dif12

Once untyped lambda calculus is put into nalebinding notation, at least in principle is possible to construct a Turing machine. In practice, it might be more feasible to directly construct one.

Anybody raising the (nalebinded) glove?

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