Abstract
In a recent paper, De Toffoli and Giardino analyzed the practice of knot theory, by focusing in particular on the use of diagrams to represent and study knots [1]. To this aim, they distinguished between illustrations and diagrams. An illustration is static; by contrast, a diagram is dynamic, that is, it is closely related to some specific inferential procedures. In the case of knot diagrams, a diagram is also a well-defined mathematical object in itself, which can be used to classify knots. The objective of the present paper is to reply to the following questions: Can the classificatory function characterizing knot diagrams be generalized to other fields of mathematics? Our hypothesis is that dynamic diagrams that are mathematical objects in themselves are used to solve classification problems. To argue in favor of our hypothesis, we will present and compare two examples of classifications involving them: (i) the classification of compact connected surfaces (orientable or not, with or without boundary) in combinatorial topology; (ii) the classification of complex semisimple Lie algebras.
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Notes
- 1.
A surface is supposed to be a connected two-dimensional topological manifold, that is it cannot be represented as the union of two disjoint nonempty open sets.
- 2.
A Riemann surface is an analytic manifold of (complex) dimension 1. From the point of view of the analysis situs, a Riemann surface is orientable.
- 3.
See for reference [2, pp. 550–551].
- 4.
[3, p. 80]. From this booklet, it does not seem that Klein has used any figure to illustrate this description.
- 5.
Note that the equivalence \(\mathbb {T}^2 \# \mathbb {P}_2(\mathbb {R}) \cong K \# \mathbb {P}_2(\mathbb {R})\) does not entail \(\mathbb {T}^2 \cong K\). In fact, \(\mathbb {T}^2\) is orientable while K is not.
- 6.
The connected sum of two surfaces \(\Sigma _1\) and \(\Sigma _2\), is obtained by deleting from \(\Sigma _1\) a disk with boundary circle \(c_1\) and from \(\Sigma _2\) a disk with boundary circle \(c_2\) before regluing \(c_1\) and \(c_2\). In particular, \(\chi (\Sigma _1 \# \Sigma _2) = \chi (\Sigma _1) + \chi (\Sigma _2) - 2\), a formula which allows inferring that \(\chi (K) = 0\), by knowing that \(\chi (\mathbb {P}_2(\mathbb {R})) = 1\).
- 7.
Dyck conjectured but did not prove that all compact surfaces can be reduced to the normal forms he had identified.
- 8.
For instance, in the case of Figs. 6 and 7, \(K = 0\) but \(r = 1\) for Fig. 6 and \(r = 0\) for Fig. 7; therefore, the two surfaces are non-homeomorphic.
- 9.
Note that the figure next to the Normalform of the Möbius band is a Möbius band cut along its middle line. We are not sure why Dyck put this figure here.
- 10.
Alexander and Brahana were both Veblen’s students at Princeton University. Alexander obtained his PhD in 1915, Brahana in 1920.
- 11.
The “connectivity” or connection order k of a compact surface S is related to its Euler characteristic \(\chi (S)\) being \(\chi (S) = 3 - k\). For the sphere \(\mathbb {S}^2\), \(\chi (S) = 2\) and \(k = 1\). In the case of non-orientable, “one-sided” in Alexander’s terms, surfaces, the \(\chi (S) = 2 - g\) and thus the genus \(g = k - 1\); in the case of orientable or “two-sided” surfaces, \(\chi (S) = 2 - 2g\) and thus \(g = (k - 1)/2\) (note that Alexander uses p instead of g).
- 12.
For an historical study on the classification of semisimple Lie algebra, see [11].
- 13.
See in particular, Elie Cartan’s doctoral Thesis [12].
- 14.
For example, let \(\mathfrak {sl}(3, \mathbb {C}) = \mathbf {A}_2\) be the vector space of \(3 \times 3\) matrices with complex coefficients of zero trace (i.e. such that the sum of the diagonal coefficients is zero), with a bracket defined by \([A, B] = AB - BA\). This would be a Lie algebra of dimension \(3^2 - 1 = 8\). The commutative subalgebra \(\mathfrak {h} \subset \mathfrak {sl}(3, \mathbb {C})\), constituted by the diagonal matrices of zero trace is a Cartan subalgebra of \(\mathfrak {sl}(3, \mathbb {C})\). The dimension of \(\mathfrak {h}\) is \(3 - 1 = 2\). The \(3^2 - 3 = 6\) roots of \(\mathfrak {sl}(3, \mathbb {C})\) (relatively to \(\mathfrak {h}\)) span a Euclidean space of dimension 2.
- 15.
- 16.
Emmy Noether, Hermann Weyl and Ernst Witt belonged to the audience of these lectures. A written version of Artin’s presentation has been recently rediscovered by Christophe Eckes and Norbert Schappacher thanks to Ina Kersten. See their commentary on the Oberwolfach Photo Collection website, https://owpdb.mfo.de/detail?photo_id=9265.
- 17.
He uses the term Veranschaulichung, which means illustration as well as visualization.
- 18.
Ch. II of his lecture is entitled Aufzählung der Diagramme (“”enumeration of the diagrams).
- 19.
The simple Lie algebra (\(\mathfrak {so}(5, \mathbb {C})\) consists of the \(5 \times 5\) antisymmetric matrices with complex coefficients.
- 20.
Similar diagrammatic representations were introduced by Coxeter [15], Witt [16] and finally Dynkin [17, 18]. However, these diagrams do not have the same semantics and correspond to different approaches in the investigation of root systems. Despite the fact that they look similar, they have a different “dynamicity”, that means one reasons differently by using them.
- 21.
See for reference [19].
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Acknowledgements
This collaboration is one of the results of the project “Les mathématiques en action” (2016–2017), funded by the Archives Henri-Poincaré and the Pôle Scientifique CLCS of the Université de Lorraine. We also acknowledge support from the ANR-DFG project FFIUM for which we thank Gerhard Heinzmann.
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Eckes, C., Giardino, V. (2018). The Classificatory Function of Diagrams: Two Examples from Mathematics. In: Chapman, P., Stapleton, G., Moktefi, A., Perez-Kriz, S., Bellucci, F. (eds) Diagrammatic Representation and Inference. Diagrams 2018. Lecture Notes in Computer Science(), vol 10871. Springer, Cham. https://doi.org/10.1007/978-3-319-91376-6_14
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