### ISTORIJA / history

 (K)NOT THEORY
Written by Chus Martínez

In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician’s knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambientisotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

Knots can be described in various ways. Given a method of description, however, there may be more than one description that represents the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram. Any given knot can be drawn in many different ways using a knot diagram. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot.

Links are also the subject of knot theory. A link consists of several knots (called components) and interwinted in any manner.

Knot theory was introduced as a scientific topic of investigation by “two middly eccentric nineteenth century Scottish physicists,” Lord Kelvin and Guthrie Tait.Today knot theory is one of the most active research areas of mathematics, but because of its developments in relation with knots and links, cibernetic studies has also included it as an useful tool. The idea of a net, as something whose nature escapes any definition other than that where elements relate to each other, so that their identity is nothing but the way in which they lead to others and the way others lead to them, was first applied by mathematicians in order to understand the complexity of certain links of knots. In that sense aiming to define any element in such a system, we would have to assume its incompleteness.

But this isn’t really new, as we know now from Pavlov’s experiments and Pudovkin’s adaptation of those for his theory of relational editing, it is clear that things are depending on what they relate to. Language is also a net of elements. Even if words are apparently defined in dictionaries these words are much older than any dictionary.  Most of the time it is not that we speak a language as much as we are spoken by that language. Of course that there is more than words in that net, there are images associated to them. (A noir, E blanc, I rouge, U vert, O bleu : voyelles. . .) The word symbol derives from the greek symbolon, an object cut in half constituting a sign of recognition when the carriers were able to reassemble the two halves. In ancient Greece, the symbolon, was a shard of pottery which was inscribed and then broken into two pieces.

Bissociation becomes here an interesting figure, a bissociation would be a relation built between elements that refer to different levels of enunciation. Bissociation has been approached many times as a creative resourse but is also the basis of misunderstanding, and of humor. Not to mention that agreements can also be the product of a bissociation. Bissociations are subject to marxist semiotics, in the sense that allows the receiver of a message to appropriate the received message and make it his own, in order to reject the author as owner and sourse of discourse inherent to speech, which is the basis of truth (as developed by Foucault) Needless to say that along this process the same words have achieved a completely different meaning.

# It is here also that Bakhtin introduces an “architectonic” or schematic model of the human psyche which consists of three components: “I-for-myself”, “I-for-the-other”, and “other-for-me”. The I-for-myself is an unreliable source of identity, and Bakhtin argues that it is the I-for-the-other through which human beings develop a sense of identity because it serves as an amalgamation of the way in which others view me. Conversely, other-for-me describes the way in which others incorporate my perceptions of them into their own identities. Identity, as Bakhtin describes it here, does not belong merely to the individual, rather it is shared by all.

But, apropriating Chus Marrtinez’s words, I must admit that the real task of a (k)not theory is to become its own failure, in the sense that the previous arguments above, only would make sense if your I-for-me arrived to the conclusion that this post is completely unreliable. And then I would agree with you because in (k)not theory between two opposite statements it is posible to include a third one that opposes each of them as the other side of a coin, even if a coin does not have three sides. Take as an example Kelvin’s words about his collaborator: “We never agreed to disent, so we always fought it out. But it was almost as great pleasure to fight with Tait as to agree with him.”

Another example is James Clerk Maxwell’s zoetrope, who influenced by the smoke box built by Tait, invented a zoetrope (an optical circular toy with an axis that when revolving in parallel to the ground, and on looking through the slits in the cylinder the figures are seen moving on the opposite side of the cylinder), but Maxwell inserted concave lenses where the slits, so that the virtual image appeared stationary as the cylinder revolved. Surplisingly when looking at the images inside the zoetrope they represent the movement of the smoke rings as described by Kelvin and Tait.

# If they have the same direction of rotation, they travel in the same direction; the foremost widens and travels more slowly, the pursuer shrinks and travels faster, till, finally if their velocities are not too different, it overtakes the first and penetrates it. Then the same goes on in the opposite order, so that the rings pass through each other alternatively.

This peculiar invention was succesfull only to the extent that it denied itself, because the next and most challenging task after demonstrating a theory is to deny it, the same way a knot is something that is done only in order to be undone at some point. The symbolon in Greece committed a similar function as it was basically a piece of clay that was broken in two in order to be joined again. In this sense (k)not theory cannot be tackled as other than the story of a great failure.