Forms and Roles of Diagrams in Knot Theory

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.

Proximal vortex cycles and vortex nerve structures. Non-concentric, nesting, possibly overlapping homology cell complexes

“…This is a form of knowledge extraction. open-3 o It is well-known that real elementary particles can have the form of knots [18], which have various forms in knot theory [45]. open-11 o Brain tissue tessellation shows an absence of canonical microcircuits [40].…”

Envisioning Transformations—The Practice of Topology

“…In knot theory, many different notations are needed and there are no 'more natural' ones. See for reference (Brown 1999) as a starting point and our previous study on knot diagrams (De Toffoli and Giardino 2014). the three elements of the mathematical practice that we have defined above and that are in our view of philosophical interest.…”

An Inquiry into the Practice of Proving in Low-Dimensional Topology

“…6 In a previous study, we considered the use of knot diagrams in relation to knot types. We claimed that the key feature of knot diagrams is their "dynamicity": experts manipulate them according to different sets of possible transformations (De Toffoli and Giardino 2014). 7 See Fomenko (1997, Ch.…”

“…(1) Knot diagrams present clear visual elements because they "intuitively" represent geometric objects, but at the same time, they allow for a syntactic control (through local moves specifically defined on them) (De Toffoli and Giardino 2014). (2) Commutative diagrams of homological algebra display a more evident syntactic component: these diagrams no longer describe geometric objects but abstract structures and relations.…”