Engaging globally with knots and riddles -- Gordian and otherwise

Engaging with Elusive Connectivity and Coherence (Part #7)

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Gordian knot: The challenge of governance, whether global or otherwise, is articulated in terms of the traditional Gordian knot with which Alexander the Great was confronted. Recent examples include:

• John Jullens: The Gordian Knot of Global Economic Growth (Strategy-Business, 15 October 2013).
• Bernard Harborne and Bernhard Metz: Cutting through the Gordian Knot: analysis of conflict and violence (World Bank, 16 June 2015)
• Gabriela Lemus: Severing the 'Gordian Knot': defying conventional wisdom and reversing current trade policy (The Huffington Post, 2 June 2015)
• Reinhard Wagner: The Gordian Knot of Global Collaboration (International Project Management Organization, 22 January 2016)
• Pawel Opala and Krzysztof Rybinski: Gordian Knots of the 21st Century (2007).

The question is whether the structure of the knot, as a challenge to comprehension, can be explored more effectively, rather than simply treating it as a metaphor for "complexity", a discussed separately (Mapping grossness: Gordian knot of governance as a Discordian mandala? 2016).

Borromean rings as braid closure: Through metaphor, related characterizations can be explored in terms of the riddle of governance and the juggling of priorities (Global Governance as a Riddle: but is a solution the answer to the question? 2018; Governance as "juggling" -- Juggling as "governance": dynamics of braiding incommensurable insights for sustainable governance, 2018).

The latter discussion notes that juggling patterns can be understood in mathematical terms in the light of the theory of braids. That theory is critical to the theory of knots and links. Artists have been inspired by the Borromean ringgs as braids (Tom Verhoeff an Koos Verhoeff, Three Families of Mitered Borromean Ring Sculptures, Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture, 2015).

Borromean rings, as a link of three unknotted loops (such that any two of the loops are unlinked) can therefore be understood as the closure of a braid, as discussed in some detail by Louis H. Kauffman (Knot Logic and Topological Quantum Computing with Majorana Fermions, arXiv, 2013).

Borromean rings as an ordered knot-set: Kauffman notes that as an ordered knot-set, the Borromean rings constitute a "scissors-paper-stone" pattern. Each component of the rings lies over one other component, in a cyclic pattern (left below).

Representations of Borromean rings (with Red surrounds blue; Blue surrounds green; Green surrounds red)
As an ordered knot-set	Toroidal version	3 Möbius strips (animation)
Reproduced from Kauffman (2013)	Reproduced from Kauffman (2006)	Video (mp4); Virtual reality (x3d; wrl)

Kauffman presented the same diagram in a communication to a less specialized audience, together with that in the centre above (The Borromean Rings: a tripartite topological relationship, Bridges, 2006). That in the centre is perhaps the most comprehensible representation of the rings. A highly provocative argument could of course be made that the link diagram (above left) may hold further significance through its correspondence to the most controversial symbol (Swastika as Dynamic Pattern Underlying Psychosocial Power Processes: implicate order of Knight's move game-playing sustaining creativity, exploitation and impunity, 2012). The Knight's move of chess, as explored, there forms a pattern curiously similar to that on the left above.

The "scissors-paper-stone" metaphor has been extended to 5-ring and 7-ring Borromean configurations (Marc Chamberland and Eugene A Herma, Rock-Paper-Scissors meets Borromean Rings, Grinnell College, 2014). As with the 3-ring game, the authors argue the case for 5-part and 7-part games with contrasting "weapons", noting that the number of such "games" has been extended to 13. Of relevance to this argument is whether the "weapons" are nations, as with the Group of 5 Permanent Members of the UN Security Council, the Five Eyes intelligence alliance, the Group of 7, or sets of mutually overriding strategic priorities of governance. Do 3-fold groups invite similar insight (Trilateral Commission, Trilateral Cooperation Secretariat)? What then of the viability of any cartel or crime ring, most notably triads?

Borromean Rings understood as weapons in a "Rock-Paper-Scissors" Game
The Game	Discordian mandala	5-fold Borromean rings	7-fold Borromean rings
Reproduced from Wikipedia		Reproduced from Chanberland and Herma (2013)

Borromean rings and "unknotting"? Such patterns merit exploration in relation to the Reidemeister moves of knot theory, as mentioned by Kauffman (2013). It is surprising to note that the relation between the Borromean rings and the Reidemeister moves has been primarily indicated through music (Robin Hayward and Christopher Williams, Reidemeister Move plays Borromean Rings, Corvo Records, 2014). This recalls the effort to use song as a mnemonic aid to comprehension of the pattern of metabolic pathways (Harold Baum, Biochemists' Song Book, 1995; The Biochemists' Songbook MP3 Files).

Interest in Borromean rings in knot theory has an intiguing relation to understanding of what is termed the unknot (namely a circle). Intertwined as they are, and as depicted, the three rings can be understood as an entanglement of unknots. However this evokes questions such as Why is the unknotting number of Borromean rings 1? (Mathematics, 29 May 2017).

To the extent that complexity may be a consequence of distorted perception of wholeness, a further question lies in the challenge of recognizing whether what is (potentially erroneously) perceived as knotted is in fact an unknot. The answer is explored in relation to Graham's number, reputed as the largest number to have ever shown up in any mathematical proof (John Baez, An Upper Bound on Reidemeister Moves, Azimuth, 9 March 2018; Alexander Coward and Marc Lackenby, An upper bound on Reidemeister moves, American Journal of Mathematics, 136, 2014, pp. 1023-1066).

Are proposals for dialogue, as is currently the case in France with respect to the Grand Débat, to be usefully reframed as an "unknotting" process? Problematic however is the charactristic of many dialogues in which the degree of knotting in fact increases rather than decreases -- despite purported aspirations to achieve an unknotted condition. Such considerations raise the question as to whether agreements may take the form of particular knots -- of which various knot tables and knot tabulation may offer an indication, notably when ordered by number of crossings and links (List of prime knots, Wikipedia).

Knot tables as an indication of conditions of dialogue and degrees of agreement?
Prime knots up to seven crossings; labeled with Alexander-Briggs notation	All simple knots admitting a projection on the plane with 9 or fewer double points
Reproduced from Wikipedia	Reproduced from knot table (Encyclopedia of Mathematics)

Of potential relevance to knotting and unknotting are explorations of geometric moves on knots, as described by Junghwan Parl (Milnor's Triple Linking Number and Derivates of Genus Three Knots, arXiv, 30 March 2016):

Double Borromean rings insertion move

Reproduced from Park (2016)

There is a further intriguing possibility that considerations of patterns of knotting may be relatted to those of patterns of knitting -- which, at least metaphorically, is valued in terms of "knitting" relations together (Evelyn Lamb, Knotted Needles Make Knitted Knots, Scientific American, 26 February 2014; Borromean Cubes, Wooly Thoughts). The feasibility of the use of the mysterious Roman dodecahedron for knitting has been variously demonstrated (Knitting with the Roman dodecahedron, YouTube, 1 July 2014; Martin Hallett, Has The Roman Dodecahedron Mystery Been Solved? YouTube, 3 June 2014). The possibility is somewhat reminiscent of the use of qhipu as the mode of communication in the Andean civilization, a pattern of knots now in process of decoding by the Harvard Khipu Database Project (String, and Knot, Theory of Inca Writing, The New York Times, 12 August 2003).

Borromean rings and quantum entanglement: The pattern above enables Kauffman to note the relation to the Greenberger-Horne-Zeilinger state, namely a certain type of entangled quantum state that involves at least three subsystems (particles). Cting P. K. Aravind (Borromean entanglement of the GHZ state, 1997), Kauffman notes:

The Borromean rings are entangled, but any two of them are unentangled. In this sense the Borromean rings are analogous to the GHZ state... Measurement in any factor of the GHZ yields an unentangled state. Aravind points out that this property is basis dependent. We point out that there are states whose entanglement after an measurement is a matter of probability (via quantum amplitudes).... New ways to use link diagrams must be invented to map the properties of such states. [emphasis in original]

The University of Bergen had a website dedicated to Radioactive Nuclear Beam Theory and the implication of the Borromean rings. The work has focused on few-body theory for light halo-like Borromean nuclei (Jan S. Vaagen, et al., Borromean Halo Nuclei: Continuum Structures and Reactions; M. V. Zhukov, et al, Bound State Properties of Borromean Halo Nuclei, 1993; C. A. Bertulani, et al, Geometry of Borromean Halo Nuclei, 2007). The 3-body quantum analogue is one where the 3-body system is bound, but none of the 2-body subsystems are bound.

Framing fundamental research in terms of "few-body" theory is suggestive of the need for a similar framing in the psychosocial sciences -- "few-category" theory or "few-concept" theory. Of potential relevance are the application of quantum insights to the social sciences by (Alexander Wendt, Quantum Mind and Social Science: unifying physical and social ontology, 2015), as discussed separately (On being "walking wave functions" in terms of quantum consciousness? 2017).

There is potentially great irony to the fundamental symbolic importance traditionally associated with the Sun, now that research is discovering the role of light halo-like Borromean nuclei in solar physics -- and more generally in astrophysics (B. V. Danilin, et al. Cluster models of light nuclei and the method of hyperspherical harmonics: successes and challenges, Physics of Atomic Nuclei, 72, 2009, 8, pp. 1272-1284).

Hyperbolic implication of Borromean rings: The complement of Borromean rings is octahedral and hence embeds geodesically in a finite volume hyperbolic four-manifold. The Borromean rings complement is tessellated into two ideal octahedra (B. Martelli:, Hyperbolic 3-manifolds that embed geodesically, arXiv:1510.06325, 2015). All known examples of geodesically bounding hyperbolic link complements are arithmetic and have each of their respective components unknotted. From that perspective the Borromean rings complement is a thrice punctured torus bundle by specifying automorphism of the surfaces in question (Christof Menzel, The Whitehead Link Complement and the Borromean Rings Complement are Torus Bundles, 11 June 1999).

Borromean rings as an Euclidean orbifold: sequence of transformations of an Euclidean cube into a hyberbolic orbifold
Adapted from D. Cooper, Orbifolds. Project Euclid, ?2000

Soap films on Borromean rings: An unusual representation of Borromean rings is based on their support for many distinctive soap films, as presented by Ken Brakke (Soap films on Borromean rings) The author indicates 14, of which a selection is presented below, although others are possible. The images were made with his Surface Evolver program. The films are noted as obeying Plateau's Laws of Soapfilms, namely that the three films meet along a curve at equal angles, and three such triple curves meet at a "tetrahedral point" at equal angles.

Selection of soap films on Borromean rings

Reproduced from website of Ken Brakke

Laws of form: Also of relevance is George Spencer-Brown's Laws of Form (1969), as variously discussed by Kauffman (Laws of Form: an exploration in mathematics and foundations). As Kauffman (2013) notes in a preamble to his discussion of Borromean rings:

Laws of Form begins with the following statement. We take as given the idea of a distinction and the idea of an indication, and that it is not possible to make an indication without drawing a distinction. We take therefore the form of distinction for the form. Then the author makes the following two statements (laws):

1. The value of a call made again is the value of the call.
2. The value of a crossing made again is not the value of the crossing

Understood in the light of the Laws of Form as constituting category boundaries, the Borromean rings (separately and together) raise the question as how such boundedness should be understood as sustaining a distinction -- and the nature of the misunderstanding when this is not appreciated. The Borromean rings are a form of (re)presentation.

As discussed separately (Form, geometry, pattern and dimensionality, 2007), the focus of Michael Schiltz (Form and Medium: a mathematical reconstruction, Image [&] Narrative, 6, 2003) follows from that of the "calculus of indications" of Laws of Form. Schiltz notes that form/medium is the image for systemic connectivity and concatenation, as described by Humberto Maturana and Francesco Varela. For Schiltz, the notion of "space" is the key to reflexivity appropriate to any discussion of form and medium, citing Spencer-Brown as follows:

In all mathematics it becomes apparent, at some stage, that we have for some time been following a rule without being aware of it. This might be described as the use of a covert convention. [â-... Its] use can be considered as the presence of an arrangement in the absence of an agreement. For example, in the statement and theorem.... it is arranged (although not agreed) that we shall write on a plane surface. If we write on the surface of a torus the theorem is not true [â-...] The fact that men have for centuries used a plane surface for writing means that, at this point in the text, both author and reader are ready to be conned into the assumption of a plane writing surface without question. But, like any other assumption, it is not unquestionable, and the fact that we can question it here means that we can question it elsewhere.

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