In mathematics, a knot is defined as a closed, non-self-intersecting curve that is embedded in three dimensions and cannot be untangled to produce a simple loop (i.e., the unknot). While in common usage, knots can be tied in string and rope such that one or more strands are left open on either side of the knot, the mathematical theory of knots terms an object of this type a “braid” rather than a knot. To a mathematician, an object is a knot only if its free ends are attached in some way so that the resulting structure consists of a single looped strand.
A knot can be generalized to a link, which is simply a knotted collection of one or more closed strands.
The study of knots and their properties is known as knot theory. Knot theorywas given its first impetus when Lord Kelvin proposed a theory that atoms were vortex loops, with different chemical elements consisting of different knotted configurations (Thompson 1867). P. G. Tait then cataloged possible knots by trial and error. Much progress has been made in the intervening years.
Schubert (1949) showed that every knot can be uniquely decomposed (up to the order in which the decomposition is performed) as a knot sum of a class of knots known as prime knots, which cannot themselves be further decomposed (Livingston 1993, p. 5; Adams 1994, pp. 8-9). Knots that can be so decomposed are then known as composite knots. The total number (prime plus composite) of distinct knots (treating mirror images as equivalent) having , 1, … crossings are 1, 0, 0, 1, 1, 2, 5, 8, 25, … (OEIS A086825).
Klein proved that knots cannot exist in an even-dimensional space . It has since been shown that a knot cannot exist in any dimension . Two distinct knots cannot have the same knot complement (Gordon and Luecke 1989), but two links can! (Adams 1994, p. 261).
Knots are most commonly cataloged based on the minimum number of crossings present (the so-called link crossing number). Thistlethwaite has used Dowker notation to enumerate the number of prime knots of up to 13 crossings, and alternating knots up to 14 crossings. In this compilation, mirror images are counted as a single knot type. Hoste et al. (1998) subsequently tabulated all prime knots up to 16 crossings. Hoste and Weeks subsequently began compiling a list of 17-crossing prime knots (Hoste et al. 1998).
Another possible representation for knots uses the braid group. A knot with crossings is a member of the braid group .
There is no general algorithm to determine if a tangled curve is a knot or if two given knots are interlocked. Haken (1961) and Hemion (1979) have given algorithms for rigorously determining if two knots are equivalent, but they are too complex to apply even in simple cases (Hoste et al. 1998).
LH Kauffman with Trefoil Knot in the back.
This slide show has been only an introduction to certain mathematical and conceptual points of view about reflexivity.
In the worlds of scientific, political and economic action these principles come into play in the way structures rise and fall in the play of realities that are created from (almost) nothing by the participants in their desire to profit, have power or even just to have clarity and understanding. Beneath the remarkable and unpredictable structures that arise from such interplay is a lambent simplicity to which we may return, as to the source of the world.
From Laws of Form and the Logic of Non-Duality
This talk will trace how a mathematics of distinction arises directly from the process of discrimination and how that language, understood rightly as an opportunity to join as well as to divide, can aid in the movement between duality and non-duality that is our heritage as human beings on this planet.The purpose of this talk is to express this language and invite your participation in it and to present the possiblity that all our resources physical, scientific, logical, intellectual, empathic are our allies in the journey to transcend separation.
Recent times have witnessed relational sociology, as arguably the major form of relational scholarship, gain considerable scholarly momentum. There is a forthcoming major handbook (Dépelteau, 2018), significant edited collections such as Conceptualizing relational sociology (Powell & Dépelteau, 2013), Applying relational sociology (Dépelteau & Powell, 2013), and in the broader leadership literatures Advancing relational leadership research (Uhl-Bien & Ospina, 2012). In addition, there have been key texts from Crossley (2011), the work of Donati (1983, 1991, 2011) has become more accessible in English (to which he thanks Margaret Archer for, stating she “greatly encouraged and assisted me in presenting my theory to an international audience (Donati, 2011, p. xvii)), and – although less engaged with by English-speaking audiences—Bajoit’s (1992) Pour une sociologie relationnelle.
The Canadian Sociological Association has established a research cluster for relational sociology, with regular symposia, meetings, and events. Significantly, in 2015 the International Review of Sociology/ Revue Internationale de Sociolgie published a special section on relational sociology. Edited by Prandini (2015) and with contributions from Crossley (2015), Dépelteau (2015), Donati (2015), and Fuhse (2015), this special section sought to ascertain whether an original and international sociological paradigm entitled “relational sociology” could be identified. Prandini (2015) argues:
A new and original social paradigm is recognizable only if it accedes to the world stage of the global scientific system constituted and structured by networks of scientific scholars, scientific contributions published in scientific journals, books, internet sites, etc., fueled by a vast array of international meetings, seminars, conferences, and so on. It is only at this global level that we can decide if a new paradigm is gaining a global stage or not. Put in other words: are we really witnessing a new and emergent sociological ‘school’, or are we observing only a sort of ‘esprit du temp’ which is able to catalyse similar intuitions and sociological insights? (pp. 1–2)
At the end of his paper, Prandini (2015) contends that there is less a paradigm (in its precise Kuhnian meaning) and instead it is better to speak of a “relational turn” in sociology. Built on a strong and clear convergence toward a common critique of classic sociological theories, it is possibly the early stages of an emerging paradigm but such a label is currently premature. The real breakthrough of this turn is in forcing social scientists to specify “accurately the ontology of society and social relation and to discover new methods and research techniques well suited to study it” (Prandini, 2015, p. 13).
Relational theory is, as Emirbayer (1997) declares, beyond any one disciplinary background, national tradition, or analytic and empirical point of view. Outside of the major centers of Europe and the USA, Yanjie Bian hosted the International Conference on Relational Sociology at the Institute for Empirical Social Science of Xi’an Jiaotong University, and Jan Fuhse hosted the international symposium Relational Sociology: Transatlantic Impulses for the Social Sciences at Humboldt University of Berlin. Donati (2011) claims that interest in social relations can be found in philosophy (from the metaphysical point of view), psychology (from the psychic point of view), economics (from the resource perspective), law (control by rule), and even biology (bioethics). The interest is also not limited to the social sciences, with Bradbury and Lichtenstein (2000) noting:
The interdependent, interrelated nature of the world has also been discovered by physicists in their study of quantum reality. In their quest to identify the basic building blocks of the natural world, quantum physicists found that atomic particles appeared more as relations than as discrete objects (Capra 1975; Wolf 1980), and that space itself is not empty but is filled with potential (Bohm 1988). Heisenberg’s discovery early this century that every observation irrevocably changes the object being observed, further fueled the recognition that human consciousness plays an irreversible role in our understanding of reality (Bachelard, 1934/1984; Wilber 1982; Jahn & Dunne 1987). (p. 552)
Apart from its widespread contemporary appeal, relational thinking has a long history. The North American stream arguably finds its roots in the New York School, European scholars such as Karl Marx, Georg Simmel, Gabriel Tarde, Norbert Elias, Niklas Luhmann, Pierre Bourdieu, Bruno Latour, among others, have long argued for various relational approaches (even if not using that label), and Emirbayer traces the tradition of privileging relations rather than substances to pre-Socratic Greek philosopher Heraclitus. What is consistently germane across these various scholars is a critique of substantialism in classic sociological accounts. This also arguably speaks to the proliferation of relational scholarship in the past few decades as globalized forces are causing a rethink of spatio-temporal conditions (e.g., the nation state and geographic borders). In breaking down the substantialist approaches, and their underlying analytical dualisms, relational scholarship asks questions of the ontological and epistemological as much as the empirical.
Contemporary thought and analysis in social theory is overrun with “turns.” In this chapter, rather than be seduced by contemporary attention to a relational turn in the social sciences, I seek to highlight some major events, trajectories, or streams of relational thought. In doing so, I am critically aware of the difficulty of arguing for relational understanding and then constructing significant events as though they are entities in and of their own right. Within the confines of a single chapter, and mindful of the role that this chapter is playing the book (e.g., setting some context/trajectory for developing my argument), my goal is to cite key developments and how they relate to one another and my argument. Given my particular interest in organizing activity, my focus is on the Human Relations Movement of the early twentieth century, the New York School of relational sociology, and then contemporary developments in sociology, leadership, and to a lesser extent, the natural sciences. While I concede that there is increasing interest in what has come to be known as “relational sociology” (see also the following chapter), relational scholarship has a long and diverse intellectual history. Importantly though, as Powell and Dépelteau (2013) note, relational sociology is not a heterogeneous label and as a collection of scholars, is still quite some way from achieving any form of consensus. Whether consensus is required, or even desirable, for relational scholarship is questionable. The diversity of ontological and methodological starting points allows scholars to investigate a wide range of phenomena. This diversity, complexity, depth, and vitality enable dialogue and debate without requiring consensus. What binds them together is their scholarly focus on relations rather than alignment with a specific empirical object and/or method of inquiry
CALL FOR PAPERS
The Relational Turn in Sociology: Implications for the Study of Society, Culture, and Persons
Special issue of the academic journal Stan Rzeczy [State of Affairs]
The relational approach, which has a long tradition, has re-emerged and strengthened, forming a new, vital movement of divergent variants in sociology. Initiated and systematically developed by Pierpaolo Donati, it has grown into what is called the Italian relational turn, later followed by a proliferation of relational sociologies of various origins, including the works of Harrison C. White, Charles Tilly, Mustafa Emirbayer, Pierre Bourdieu and others. After the postmodern diffusion and beyond the stagnation of interpretative against normative conceptualizations of social life, relational sociology offers new conceptual tools and plays a leading role in reconstructing sociology both on theoretical and applied planes.
Modern sciences are founded on the study of relations, rather than essences or substances. From the outset, the relational approach has had to pave its way in sociology against holistic (“science of society”) and nominalistic (“science of individuals”) orientations. Social relations are among the key sociological concepts and have been studied as constitutive for social bonding. On the micro-level, interpersonal relations have been in the center of attention in the area where sociology and social psychology overlap. The relational turn consists not only of focusing on social relations; it also involves introducing relational categories of analysis.
The category of social relations is certainly not new in social theory. What is new is the way of looking at them. Contemporary relational thinking assumes radical changes in the ontological, epistemological, and phenomenological status of social relations. Refocusing on social relations, on their constitution and emergent effects leads us to a new way of describing, understanding and explaining social and cultural phenomena as relational facts.
A particularly significant feature of relational sociology resides in its capacity to broaden the theory of the human subject not only as a self, agent, and actor, but also through the development of the concept of the person; more precisely, through deeper research on the relational constitution of the human person as a social subject emerging from relational reflexivity (dialogue between ‘I’, ‘Me’, ‘We’, ‘You’ in a situated social context) – in other words, a view of the human person as homo relatus. Analyzing these processes leads to a sui generis relational theory of agency.
Various or divergent theories of contemporary social and cultural processes evoke relationality, but relational analysis differs from “relationistic” positions. Most existing approaches, both historical and modern, cannot be considered relational sociology in a true sense unless the social relation is conceived as a reality sui generis and society is conceptualized as a network of social relations.
“Turn” refers to a gradual transformation of the field of scientific theories, rather than to a scientific revolution. Several characteristic features of a “turn” appear to correspond well with significant traits of the relational turn: an epistemological rupture, which is brought about by introducing an innovative vocabulary that opens up new analytic perspectives; an attempt to reconstruct the scientific domains of knowledge under conditions of their growing fragmentation; introduction of a novel perspective that shows existing knowledge in a new light; moving on from the research object to the category of analysis. These are the features of a genuine new intellectual movement that enters into debates and polemics, particularly as regards various ways of understanding relations and relationality.
The synergetic effect of a creative exchange of ideas between the founders of theories that have been independently pursued – the relational theory of society developed by Pierpaolo Donati and the theory of morphogenic society, developed on the basis of critical realism by Margaret S. Archer – proves particularly fruitful for the study of the after-modern and the new possibilities of a morphogenic society, in which the challenge of re-articulating social relations remains of central importance.
The aim of this special issue is to reflect upon the innovative potential of contemporary relational theorizing of society, culture, and persons and to go beyond superficial statements on relational sociology by addressing these issues through in-depth investigations. We invite authors to take on problems of relational sociology by discussing its main assumptions, by conceptual clarifications, by re-articulating the concepts pertinent to understanding social phenomena in relational terms, and by empirical studies guided by methodological rules of relational analysis.