Abstract
The aim of this article is to explain why knot diagrams are an effective notation in topology. Their cognitive features and epistemic roles will be assessed. First, it will be argued that different interpretations of a figure give rise to different diagrams and as a consequence various levels of representation for knots will be identified. Second, it will be shown that knot diagrams are dynamic by pointing at the moves which are commonly applied to them. For this reason, experts must develop a specific form of enhanced manipulative imagination, in order to draw inferences from knot diagrams by performing epistemic actions. Moreover, it will be argued that knot diagrams not only can promote discovery, but also provide evidence. This case study is an experimentation ground to evaluate the role of space and action in making inferences by reasoning diagrammatically.
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Notes
Kirsh and Maglio distinguish between pragmatic actions, i.e.“actions performed to bring one physically closer to a goal”, and epistemic actions, i.e. “actions performed to uncover information that is hidden or hard to compute mentally”, by examining their role in Tetris, a real-time, interactive video game.
We are developing an account of the peculiarities of the practice of low-dimensional topology with particular focus on proving, using different kind of diagrams and visual material in general (De Toffoli and Giardino, forthcoming).
The vast majority of knot theory only deals with tame knots, i.e. knots that admit a diagram with only a finite number of intersection points. This restriction is meant to ban so-called wild knots, which are “monsters” in Lakatos’ terminology. Every tame knot is equivalent to a smooth knot, that is why it is common to consider only smooth knots or other equivalent categories. A simple curve is a curve without self-intersections.
Two knots K 1 and K 2 are ambient isotopic if there exists a continuous map: \({h: {\mathbb{R}}^3 \times [0, 1] \rightarrow {\mathbb{R}}^3, }\) with h t (x) : = h(x, t), such that (i) \({h_t: {\mathbb{R}}^3 \rightarrow {\mathbb{R}}^3}\) is a homeomorphism for all t, (ii) \(h_0=\mathrm{id}\) and (iii) h 1(K 1) = K 2. Ambient isotopies model the deformations on knots that we can perform without cutting and then pasting the two cut ends.
It is also possible to partially translate diagrams and moves on them into codes (Adams 1994, Ch. 2). Many of such codes (like the Dowker Code) have been developed to the aim of using computers in order to classify knots. However, the possibility of translating a knot type or diagram into a code, that is their potential inter-translatability, does not tell us anything about the way in which diagrams are interpreted and effectively used in the practice of knot theory.
The moves presented here are basic in knot theory. However, other moves can be defined for more specific aims. See for example the Kirby calculus for surgery equivalences (Kirby 1978).
Let X be a topological space and ∼ an equivalence relation on it. The quotient space Y = X / ∼ is defined to be the set of equivalence classes of elements of X.
The crossing number of a knot diagram is the number of its crossings. Let \({{\mathcal{K}}}\) be a knot type. The crossing number \({C({\mathcal{K}})}\) of \({{\mathcal{K}}}\) is the minimum over the crossing numbers of all the diagrams representing it. A minimal diagram is a diagram presenting \({C({\mathcal{K}})}\) crossings.
Bleiler [1984] proved that the unknotting number of a knot type is not necessarily appreciable from a minimal diagram of it. In this specific case, he proved that the unknotting number of the knot type 108 is exactly two.
Shin and Lemon use this term to refer to Euler’s belief that the same kind of visual containment relation among areas used in the case of two universal statements can be used as well in the case of two existential statements; this is not correct and the employed representation raises a “damaging ambiguity” (Shin and Lemon 2008).
See Lickorish (1997, Ch. 11) for a detailed explanation.
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Acknowledgments
We wish to thank Francesco Berto and Achille Varzi for having made our collaboration possible. We presented our work on knot diagrams in various occasions and received helpful feedback. In particular, we are thankful to Anouk Barberousse, Roberto Casati, José Ferreiros, Marcus Giaquinto, Hannes Leitgeb, Øystein Linnebo, John Mumma, Marco Panza, and John Sullivan. We thank three anonymous referees for their thoughtful comments and suggestions that helped us improve a first version of the article. Silvia De Toffoli acknowledges support from the Berlin Mathematical School and from the DFG Collaborative Research Center SFB/TR 109 “Discretization in Geometry and Dynamics”. Valeria Giardino is grateful to the Spanish Ministry of Education and to the Exzellenzcluster 264 - TOPOI for supporting her research.
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De Toffoli, S., Giardino, V. Forms and Roles of Diagrams in Knot Theory. Erkenn 79, 829–842 (2014). https://doi.org/10.1007/s10670-013-9568-7
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DOI: https://doi.org/10.1007/s10670-013-9568-7