, Volume 79, Issue 4, pp 829–842 | Cite as

Forms and Roles of Diagrams in Knot Theory

Original Article


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.


Concrete Object Epistemic Action Reidemeister Move Algebraic Reasoning Diagrammatic Reasoning 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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.


  1. Adams, C. (1994). The knot book. New York: Freeman.Google Scholar
  2. Bleiler, S. A. (1984). A note on unknotting number. Mathematical Proceedings of the Cambridge Philosophical Society, 96(3), 469–471.CrossRefGoogle Scholar
  3. Brown, J. R. (1999). Philosophy of mathematics: An introduction to the world of proofs and pictures. New York: Routledge.Google Scholar
  4. Colyvan, M. (2012). An introduction to the philosophy of mathematics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  5. Cromwell, P. (2004). Knots and Links. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  6. De Cruz, H., & De Smedt, J. (2013). Mathematical symbols as epistemic actions. Synthese, 190(1), 3–19.Google Scholar
  7. De Toffoli, S., & Giardino, V. (forthcoming). An inquiry into the practice of proving in low-dimensional topology, Boston Studies in the Philosophy of Science.Google Scholar
  8. Giaquinto, M. (2007). Visual thinking in mathematics. Oxford: Oxford University Press.CrossRefGoogle Scholar
  9. Giardino, V. (2013). A practice-based approach to diagrams. In A. Moktefi, S.-J. Shin (Eds.), Visual reasoning with diagrams, studies in universal logic (pp. 135–151). Birkhauser: Springer.Google Scholar
  10. Kirby, R. (1978). A calculus for framed links in \({{\mathbb{S}}^3}\). Inventiones Mathematicae, 43, 35–56.CrossRefGoogle Scholar
  11. Kirsh, D., & Maglio, P. (1994). On distinguishing epistemic from pragmatic action. Cognitive Science, 18, 513–549.CrossRefGoogle Scholar
  12. Lickorish, R. (1997). An introduction to knot theory (graduate texts in mathematics). New York: Springer.CrossRefGoogle Scholar
  13. Macbeth, D. (2012). Seeing how it goes: Paper-and-pencil reasoning in mathematical practice. Philosophia Mathematica, 20, 58–85.CrossRefGoogle Scholar
  14. Mancosu, P. (Eds.) (2008). The philosophy of mathematical practice. Oxford: Oxford University Press.Google Scholar
  15. Muntersbjorn, M. M. (2003). Representational innovation and mathematical ontology. Synthese, 134, 159–180.CrossRefGoogle Scholar
  16. Shin, S.-J., & Lemon, O. (2008). Diagrams. Entry in the Stanford Encyclopedia of Philosophy. Retrieved 2012, from

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.Institut Jean Nicod, CNRS-EHESS-ENSParisFrance

Personalised recommendations