Forms and Roles of Diagrams in Knot Theory Silvia De Toffoli and Valeria Giardino

I made a lot of use of this in my MRes work, but never wrote up any notes on it, so here goes.

As Mancosu (2008, p. 2) summarizes, ‘‘the epistemology of mathematics needs to be
extended well beyond its present confines to address epistemological issues having to
do with fruitfulness, evidence, visualization, diagrammatic reasoning, understanding,
explanation and other aspects of mathematical epistemology which are orthogonal to
the problem of access to ‘abstract objects’.’’ p. 829

A figure per se does not have a meaning; in order to become
meaningful, it has to be considered inside a particular context of use, and therefore
interpreted in such a context. It is only when the intention behind the figure is
recognized that the figure is seen as a representation and as a consequence it becomes
an illustration or a diagram. By illustration, we mean a static representation, which
can be useful by conveying information in a single display, but where modifications
are not well-defined. By diagram, we mean a dynamic representation, on which we
can perform moves that can count as inferential procedures. Diagrams are dynamic
inferential tools that are modified and reproduced by the experts for various epistemic
purposes. They do not only represent strategies to solve problems but also give
evidence for their solutions. p. 830

Knot diagrams are in a sense privileged points of view on knots and knot types: They
display only a certain number of properties by selecting the relevant ones. If illustrations
are analogous to pictures, diagrams are like maps. In order to draw a map, it is required
to define conventions that would make it legible. p. 833

Moreover, there are particular geometric properties, such as the
presence of symmetries, which can be used to infer properties of the represented knot
type. In the same example, the diagrams in Fig. 3a and in Fig. 3b, which are the same
diagram also if considered geometrically, display a clear symmetry of order four,
while the diagram in Fig. 3c does not. p. 834

Experts perform
actions on diagrams by re-drawing them in appropriate ways, according to the way
they interpret them. For this reason, novices need to train their imagination in order to
recognize the various possible moves on diagrams, and then be able to effectively use
them. Moreover, these manipulations are similar to the manipulations we can perform
on concrete objects, but instead of having a pragmatic aim, they have an epistemic one
(Kirsh and Maglio 1994). The use of diagrams triggers a form of manipulative
imagination that gets enhanced by the practice. p. 836

This is something that Giardino expanded on in her talk at Stanford – this manipulative imagination and some way of defining that idea. It was a very interesting talk, making use of Walton’s theory of art and make-believe alongside some embodied mind ideas.

The meaning of a knot diagram is fixed by its context of use: Diagrams are the
results of the interpretation of a figure, depending on the moves that are allowed on
them and at the same time on the space in which they are embedded. Once we
establish the appropriate moves, we fix the ambient space, thus determining the
different equivalence relations. The context of use does not have to be pre-defined.
This is not a ‘‘damaging ambiguity’’11; on the contrary, it expresses the richness ofthis notation and explains why it is effective in promoting inference. The
indetermination of meaning makes different interpretations available, and therefore
allows attending to various properties and moves. In order to ‘see’ a diagram in a
figure we have to recognize the relevant information and be aware of the possible
modifications. For this reason, the effectiveness of a diagram increases with
expertise: Only experts are able to fully exploit the richness of the different possible
meanings that a figure can acquire. p. 838-839

The knot group can be used to distinguish knots and
in particular to give an alternative proof of the existence of non-trivial knots.
Moreover, it shows that diagrammatic and algebraic reasoning can be related since
knot diagrams can be interpreted algebraically. This reveals that these diagrams can
be used as syntactic devices. p. 841

 

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