Forms and Roles of Diagrams in Knot Theory
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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.
KeywordsConcrete Object Epistemic Action Reidemeister Move Algebraic Reasoning Diagrammatic Reasoning
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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