Ecological hyperknots

An ecological hyperknot is a way of investigating distinguishable recursive organisations that condition one another’s continuation. A lesson, for example, brings together embodied participation, experience, communication, computation and organisational decisions. These relations can enable further activity without becoming one operation, sharing one purpose or forming a single self-producing super-system. (Watson and Brezovec, 2026, AE2.6)

Why use the image of a knot?

The image directs attention to connections that matter to continuation. A timetable helps make a class possible. Classroom activity produces attendance records and assessments that condition later timetabling and support. An AI tool changes the work needed for feedback, while that feedback can change what is supplied to later tool use. The resulting relations have direction, timing and unequal consequences.

Calling this a hyperknot is a proposal about what to investigate. The analyst must still identify the relevant operations, distinguish productive relations from incidental contact, and explain what would change if a relation were interrupted. The image has little value when it merely decorates the statement that everything is connected.

Recursive form and mathematical sources

Kauffman’s work on eigenforms develops the relation between recursive activity and the forms an observer recognises. His “Reflexivity and Eigenform” discusses von Foerster’s treatment of objects as tokens for recurring behaviours. This offers a way of investigating apparent stability through the operations that constitute it. It does not by itself provide evidence for the stability of a particular institution or technology. (Kauffman, 2009)

Watson’s 2026 hyperknot work extends the inquiry towards heterogeneous relations of continuation. The current AE manuscript also engages mathematical work on linkoids and open curves. Barkataki and Panagiotou, and subsequently Barkataki, Kauffman and Panagiotou, develop specified mathematical objects and constructions. Their work provides a discipline of definition and transformation; its applicability to an empirical ecological diagram has to be established separately. (Watson, 2026, Hyper-Knots) (Barkataki and Panagiotou, 2022) (Barkataki, Kauffman and Panagiotou, 2024) (Watson and Brezovec, 2026, AE2.6)

Three things to distinguish

• The target: the actual configuration being investigated, such as an assessment practice over one term.
• The model: the selected operations, relations, boundaries and timescales used to explain that target.
• The diagram: a drawing that displays the model under stated conventions.

A line crossing another line on a page does not establish an interaction. Rearranging the drawing need not change the model. Changing the model need not change the actual practice. A mathematical invariant requires a defined domain, permitted transformations and proof of preservation. Visual similarity or repeated outcomes cannot substitute for these requirements. (Watson and Brezovec, 2026, AE2.6)

A worked example

Return to the constructed case in which corrected feedback fails to amend a learner profile. One continuation is the classroom dialogue through which the student’s explanation is understood. Another is the sequence of authorised assessments. A third concerns the technical storage and retrieval of records. The same expression can participate in each, acquiring different significance and authority. (Watson, 2026, Operability)

The inquiry asks how these continuations condition one another. A teacher’s judgement may change the record; the record may direct the next lesson; that lesson may create an opportunity for the student to challenge the judgement. Alternatively, the challenge may never reach the record. The difference lies in a specific relation that can be investigated, not in the mere presence of several actors.

Now extend the interval. Teachers may repeatedly reconcile mismatched records by working additional hours. The service continues to produce apparently successful outputs, but its continuity draws on time that would otherwise sustain dialogue or rest. An alternative arrangement might reduce routine work and increase those capacities. Both possibilities belong in the inquiry. They require evidence about work, timing and subsequent participation.

Relation to infrastructure and enactive research

Infrastructure research already examines the situated relations that make technical activity possible. Participatory sense-making, developed by De Jaegher and Di Paolo, investigates how interaction and individual activity jointly shape meaning. These are substantive neighbouring accounts, each with its own commitments. The hyperknot proposal connects distinct continuations while asking which relations renew or impair their conditions; it should be evaluated against what those existing accounts can explain. (Star and Ruhleder, 1996) (De Jaegher and Di Paolo, 2007)

Questions and limitations

Which capacities are continuing, and for whom? Which relations renew their conditions, which merely transmit an effect, and which transfer a burden? Can an alternative model explain the outcome more simply? What evidence would show that an apparently essential relation is replaceable?

A hyperknot has no automatic collective measure of success. An institution’s reporting continuity can coexist with declining opportunities for participation. Nor does causal interdependence distribute responsibility equally: a person supplying compensatory work may have little control over the procedure. Ethical assessment requires explicit reasons about harm, authority, participation and remedy. (Watson and Brezovec, 2026, AE2.6)

Related entries

Rough closure identifies consequentially unsettled relations. Semantic transduction examines changes across settings. The history of AE places hyperknots alongside related scholarly debates.

Sources support the particular claims discussed; their inclusion does not imply endorsement of AE. Publication and manuscript status are identified below.

Watson and Brezovec, 2026, AE2.6

Watson, S., and Brezovec, E. (2026). Autopoietic Ecology: Rethinking Systems, Meaning, and Matter. AE2.6, revised manuscript dated 15 September 2026. Unpublished manuscript. Relevant locators: preface; chapters 3–6, 10–12 and 15–18; appendices A–C.

Kauffman, 2009

Kauffman, L. H. (2009). Reflexivity and Eigenform. Constructivist Foundations, 4(3), 121–137. Author-hosted full text. Source.

Watson, 2026, Hyper-Knots

Watson, S. (2026). Ecological Hyper-Knots and Eigenforms: A Topological Generalization of Autopoietic Operations. March. Public preprint. Later AE2.6 qualifications are distinguished from this earlier formulation. Source.

Barkataki and Panagiotou, 2022

Barkataki, K., and Panagiotou, E. (2022). The Jones polynomial of collections of open curves in 3-space. Proceedings of the Royal Society A, 478(2267), 20220302. Source.

Barkataki, Kauffman and Panagiotou, 2024

Barkataki, K., Kauffman, L. H., and Panagiotou, E. (2024). The virtual spectrum of linkoids and open curves in 3-space. Journal of Knot Theory and Its Ramifications, 33(3), 2450006. Source.

Watson, 2026, Operability

Watson, S. (2026). Generative AI and the recursive constitution of operability: Rough closure, semantic transduction and ecological hyperknots. Cambridge EdgeLab working paper, version 1, 18 September. Manuscript consulted; not peer reviewed. Constructed case and prospective research propositions.

Star and Ruhleder, 1996

Star, S. L., and Ruhleder, K. (1996). Steps toward an ecology of infrastructure: Design and access for large information spaces. Information Systems Research, 7(1), 111–134. Source.

De Jaegher and Di Paolo, 2007

De Jaegher, H., and Di Paolo, E. (2007). Participatory sense-making: An enactive approach to social cognition. Phenomenology and the Cognitive Sciences, 6, 485–507. Source.