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How to notate a crossing of strings? On Modesto Dedò’s notation of braids

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Abstract

As is well known, it was only in 1926 that a comprehensive mathematical theory of braids was published—that of Emil Artin. That said, braids had been researched mathematically before Artin’s treatment: Alexandre Theophile Vandermonde, Carl Friedrich Gauß and Peter Guthrie Tait had all attempted to introduce notations for braids. Nevertheless, it was only Artin’s approach that proved to be successful. Though the historical reasons for the success of Artin’s approach are known, a question arises as to whether other approaches to deal with braids existed, approaches that were developed after Artin’s article and were essentially different from his approach. The answer, as will be shown, is positive: Modesto Dedò developed in 1950 another notation for braids, though one, which was afterward forgotten or ignored. This raises a more general question: what was the role of Artin’s notation, or, respectively, Dedò’s, that enabled either the acceptance or the neglect of their theories? More philosophically, can notation be an epistemic technique, prompting new discoveries, or rather, can it also operate an as obstacle? The paper will analyze the method introduced by Dedò to notate braids, and also its history and implications. It aims to show that Dedò, in contrast to Artin, focused on factorizations of braids and the algebraic relations between the operations done on these factorizations. Dedò’s research was done against the background of Oscar Chisini’s research of algebraic curves on the one hand and of Artin’s successful notation of braids on the other hand. Taking this into account, the paper will in addition look into the epistemic role of notation, comparing Dedò’s work with Artin’s, as both presented different notations of braids and their deformations.

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Notes

  1. It is noticeable that with respect to weaving, Grünbaum and Shephard in their paper “Satins and Twills: An Introduction to the Geometry of Fabrics” noted in 1980 that the “literature [on weaving] is almost entirely concerned with the practical aspects of weaving; any treatment of the theoretical problem of designing fabrics with prescribed mathematical properties is conspicuously absent. And this is so in spite of the fact that many fabrics which are mathematically interesting were discovered empirically long ago by practitioners of the weaver's craft. One wonders how geometers can fail to be fascinated by the diagrams of fabrics that abound in the literature” (Grünbaum and Shephard 1980, p. 139). Cf. also Harlizius-Klück (2004).

  2. “L’ouvrier qui fait une tresse, un réseau, des noeuds, ne les conçoit pas par les rapports de grandeur, mais par ceux de situation […], c'est l'ordre dans lequel sont entrelacés les fils. II seroit donc utile d'avoir un système de calcul […] une notation qui ne représentât que l'idée qu'il se forme de son ouvrage […]. Mon objet ici n'est que de faire entrevoir la possibilité d'une pareille notation, & son usage dans les questions sur des tissus de fils.”

  3. For a detailed treatment of the research of the above-mentioned mathematicians, see Epple (1998, 1999) and Höppner (2007).

  4. With this definition, one considers also singular points. An example would be the curve y2 = x of degree 2; the point x = 0 is a branch point with respect to the projection p, since it has only one preimage. Other examples would be the curves y2 = x2 or y2 = x3, which have a node resp. a cusp.

  5. The parameter t corresponds to the function describing the loop: \( t : [0,1] \to x \)-axis; hence, Chisini considers t as ranging from 0 to 1.

  6. “La tresse algébrique diffère des tresses de M. Artin en ce sens que la tresse algébrique est naturellement partagée en secteurs […], chacun correspondant a un lacet […].”

  7. “[…] dalle variazioni per continuità che danno sistemi di cappi equivalent.”

  8. Chisini (1933, p. 1151): “The model that we have indicated as the simplest and most symmetrical […] corresponds to the case that the point O is very close to the branch point.” [“Il modello che abbiamo indicato come il piu semplice e simmetrico […] corrisponde al caso che il punto O sia molto vicino al punto di diramazione”].

  9. Private communication with Piera Manara, e-mail from February 21, 2018. See also Fig. 14 for the printing plates of figures, which may resemble Dedò's models.

  10. Private communication with Maria Dedò, e-mail from February 27, 2018.

  11. “Avevamo costruito […] un gran numero di modelli di superficie […] e questi modelli avevamo distribuito […] in due vetrine. Una conteneva le superficie regolari per le quali tutto procedeva come nel migliore dei mondi possibili […]. Ma quando cercavamo di verificare queste proprietà sulle superficie dell'altra vetrina, le irregolari, cominciavano i guai e si presentavano eccezioni di ogni specie. Alla fine lo studio assiduo dei nostri modelli ci aveva condotto a divinare alcune proprietà che dovevano sussistere, con modificazioni opportune, per le superficie di ambedue le vetrine; mettevamo poi a cimento queste proprietà con la costruzione di nuovi modelli. Se resistevano alla prova, ne cercavamo, ultima fase, la giustificazione logica. Col detto procedimento, che assomiglia a quello tenuto nelle scienze sperimentali, siamo riusciti a stabilire alcuni caratteri distintivi tra le famiglie di superficie.”

  12. Most of the models were bought in Germany and brought to Italy; Cf. Palladino and Palladino (2009) and Giacardi (2015). However, see also Friedman (2018, pp. 141–150) for a detailed treatment of the cardboard models of the pseudosphere Beltrami at the end of the 1860s.

  13. “Tale passaggio si può fare operando direttamente su un modello materiale, ma si incontrano notevoli difficoltà anche perché il modello non registra la storia dei vari passaggi intermedi e, quindi, se si commettono errori (il che troppo spesso succede) bisogna ricominciare da capo.

    Scopo di questa nota e quello di fondare un'algebra che sostituisca il maneggio di questi modelli.”

  14. “E pertanto ovvio che si potrà, in vari modi, simbolizzare un tratto comunque complesso, corrispondente ad un cappio che vada ad un punto cli diramazione semplice o, anche, ad un qualsiasi punto critico. Conseguentemente diverrà possibile rappresentare qualsiasi treccia algebrica e si potrà quindi tentare di istituire l’‘algebra’ generale di tali trecce.”

  15. “Questa proposizione, per sua natura, si può dimostrare mediante una verifica sperimentale; questa verifica ho eseguito, ed è facilmente ripetibile, sul modello materiale: ed è anche facilmente immaginabile osservando le semplici figs. 8, 9.”

  16. “Lo trecce algebriche sono trecce particolari e pertanto dalla trattazione di ARTIN esulano le proposizioni che per noi hanno particolare interesse.” Recall also Chisini’s short argument in 1952, why his braids differ from Artin’s.

  17. “Anche qui è immediata la verifica […] sul modello materiale, o anche sulle fig. […]”.

  18. “Qui si noti che ogni cambiamento del sistema di cappi si ottiene mediante scambi di due cappi consecutivi, e che per un tale scambio si presentano due casi (come è noto) secondo che si lasci fermo il primo e si faccia passare il secondo dalla sinistra alla destra di questo, o, viceversa, si lasci fermo il secondo e si faccia passare il primo dalla destra alla sinistra di questo.”

  19. “Ci occorrerà spesso di operare con le P ed S sui due tratti consecutivi a e b: l' operazione P verrà indicata [as above] […] La notazione precedente ci permetterà pure di indicare una successione di operazioni P e S. Cosi ad esempio la scrittura […].”

  20. Recall that Chisini notated the first term in the product as a + b + a−1.

  21. “Come si vedrà è possibile effettuare questo passaggio in modo puramente formale, senza ricorrere all'aiuto di modelli materiali e senza invocare proprietà della, curva algebrica rappresentata.”

  22. Recall that 1 2 1 3 = 2 3 1 2.

  23. Later Dedò will denote the appearance of two identical consecutive braids: a a as a2, and the appearance of three identical consecutive braids: a a a as a3. a2 corresponds to a node (as was noted above), whereas a3 corresponds to a cusp. Note that Dedò implicitly understands here that the notated braids belong to an algebraic structure together with an associated multiplication; for example, the notation 1 23, which is associated with cusp, stands for 1 2 1 2 1 2; the usage of the exponent 3 refers to the ordinary usage of elements in an algebraic structure, although Dedò does not prove that braids may be multiplied with one another (but only refers to Artin’s work by passing).

  24. If i < k < j, then the obtained braid would be a canonical one, denoted as i j.

  25. However, the sextic curve with six cusps not on a conic is not even mentioned, and its associated characteristic braid is therefore not calculated.

  26. Indeed, the curve of degree 4 with three cusps is also a branch curve, as Dedò notes (Dedò 1950, p. 250).

  27. Artin also returned to work on the braid group between 1947 and 1950; see Artin (1947, 1950).

  28. One should also recall that the annual meeting of the German Mathematicians’ Association (Deutschen Mathematiker-Vereinigung; DMV) took place in 1924 in Innsbruck from 21 to 27 September as part of the 88th Assembly of the Society of German Natural Scientists and Physicians (Gesellschaft Deutscher Naturforscher und Ärzte). Artin was among a total of 50 speakers and he gave there a talk, entitled “The Braid Problem” (“Das Zopfproblem”; See: DMV 1925, S. 81). The report of DMV also notes that “an overview of the lecture could not be obtained from the lecturer” [“Eine Übersicht des Vortrages war von dem Vortragenden nicht zu erlangen”]. The other speakers in the session were Hellmuth Kneser, Robert Furch, Karl Menger and Arthur Moritz Schoenflies (ibid., pp. 81–83).

  29. The joint research of Artin and Schreier on braids ended in 1926, and Artin speaks of this research in this year as “going astray topologically” (Frei and Roquette 2008, p. 95). This expression is to be found in a letter from Artin to Helmut Hasse, written on the February 10, 1926.

  30. This is not to say that Artin did not think of attempting to deal with this type of “weaving” mathematically. Examining more closely the work of Schreier, Artin’s collaborator in the paper, one sees that the two already thought about this problem in 1924. Schreier, in a letter from August 29, 1924, to Karl Menger, wrote the following: “Artin’s guiding thought was to first pick out a class of objects that are already sufficiently general, in order to bring it closer to knot theory, yet simple enough to be treatable. Hence, the condition of the ‘non-returning-backwards’ [braid]. Of course, he also wants to investigate these braids later on.” (Odefey 2014, p. 72) [“Der leitende Gedanke von Artin war doch gerade, zunächst eine Klasse von Objekten herauszugreifen, die schon hinlänglich allgemein ist, um den Knoten an den Leib zu rücken, und dennoch einfach genug, um behandelbar zu sein. Daher die Bedingung des ‘Nichtzurücklaufens’. Natürlich will er später auch zurücklaufende Zöpfe untersuchen.”] Thus, already in 1924, Artin and Schreier made a conscious decision not to consider braids which “return backwards”, and neither of them reconsidered the subject after 1926. The reference to knot theory is important, since Artin in his 1926 paper mentioned Alexander’s theorem from 1923, that every knot or link can be presented as a closure of a braid (see Alexander 1923). This means that such returning-backwards braids cannot even occur if one looks on braids obtained from knots (or links).

  31. Moreover, the way Artin notated the generators and relations did not only enable a comfortable presentation of the group, but also paved the way to the theorem, that the braid group can be also presented as a group produced with only two generators. Working with the braid group Bn with n strings, Artin denotes a = σ1σ2σ3 … σn−1 and σ = σ1. He proves, using the above relations, and without any diagrams or diagrammatic reasoning, that the braid group Bn can be presented as generated by a and σ with the relations: “an = (aσ)n−1 and σ ⇄ aiσ a−I for 2 ≤ i ≤ n/2.” (Artin 1926, p. 54) The notation ⇄ denotes commutativity between two elements in a group and was introduced in 1924 by Nielsen (1924, p. 173), though it was not used often during the following years in the context of braid theory. Note that Nielsen presents a plurality of notations in the context group theory, although most of them are not used these days. However, the investigation of the epistemic power of these notations is outside the scope of this paper.

  32. Artin noted several times this mapping; see for example: Artin 1926, pp. 54–55.

  33. Recall here Castelnuouvo and Enriques’ “experimental” approach while working with material models.

  34. Recall that 1 4 in Artin’s notation equals to \( \sigma_{1}^{ - 1} \sigma_{3} \sigma_{2}^{ - 1} \sigma_{3}^{ - 1} \sigma_{1} \), while 2 3 equals to \( \sigma_{2}^{ - 1} \).

  35. For example, with the mathematicians Werner Burau and W. Fröhlich (see Friedman 2019, pp. 48–51).

  36. “Aus Deinen Zeichnungen und Bemerkungen geht nicht klar hervor, wie Du Zopf und Komposition zweier Zöpfe präzise definierst […] Was ein Beispiel für die Nichtvertauschbarkeit von gewöhnt. Zöpfen betrifft, so ist Dein Beispiel (\( \sigma_{1} \sigma_{2}^{2} \sigma_{1} \ne \sigma_{1}^{2} \sigma_{2}^{2} \)) natürlich in Ordnung […] Beliebig viele Beispiele erhältst Du aus der (nicht-trivialen) Bemerkung, daß zwischen den Zöpfen \( A = \sigma_{1}^{2} \) und \( B = \sigma_{2}^{2} \)keinerlei Relation besteht, daß also die Gruppe {A, B} frei ist. Daher z.B. AB ≠ BA etc.” (underline in the original) As can be seen also from Schreier’s letter, it was also common to accompany discussions on braids with figures and drawings, though from Schreier’s response to Menger one can already observe a preference toward algebraic reasoning, which was accompanied by the notation. (Indeed, Schreier does not accompany his answer with any drawing.)

  37. “La conoscenza delle trecce delle curve di diramazione (trecce di diramazione) risulta importante perché la treccia di una curva C, sui cui fili siano deposti gli scambi relativi alle determinazioni di un piano multiplo (diramato da C), rappresenta in modo univoco tutta la famiglia di superficie F, birazionalmente identiche, che danno lo stesso piano multiplo e indica chiaramente di queste F molte proprietà algebriche e topologiche. Se poi la C è di diramazione per un solo piano multiplo (ciò che presumibilmente accade per n> 4, ed è comunque facilmente verificabile sulla treccia) la sola treccia di C basta a rappresentare la suddetta famiglia di superficie F. E qui importa osservare che, in particolare, se la treccia deriva per continuità (con operazioni P ed S) da una diramante un piano multiplo generale di ordine n (cioè rappresentante una superficie di ordine n) fra le precedenti superficie F ve ne è anche una di ordine n.”

  38. “Infatti associamo […] alle quattro determinazioni […] quattro cappi successivi rettilinei […], e ad essi leghiamo i quattro scambi (il cui prodotto è l'identità). […] Possiamo cosi verificare che sono soddisfatte le condizioni di invarianza di ENRIQUES e cioè verificare che […].”

  39. “Data una curva algebrica è praticamente impossibile costruirne la treccia caratteristica […]. Pertanto hanno interesse tutte le costruzioni a priori (indirette) di trecce algebriche […].”

  40. This technique was employed while researching branch curves of algebraic complex surfaces; the deformation of these curves was obtained from a deformation of a family of surfaces degenerated into a simpler surface (e.g., see Chisini 1936, 1952a).

  41. “Notiamo però che procedimenti topologici di questa natura hanno sempre condotto a trecce che sono state verificate effettivamente esistenti come trecce algebriche; cosi questo procedimento potrebbe essere considerato come un postulato costruttivo.”

  42. Cf. footnote 23.

  43. If d is the number of nodes, c the number of cusps and n the degree of the curves, then the condition is \( d + 2c + 3 < n( n + 3)/2. \)

  44. Moishezon noted the following: “The parameter space of degree m algebraic curves in \( {\mathbb{P}}^{2} \) has a finite stratification such that for any two curves S1, S2 of the same stratum the corresponding embeddings […] are topologically equivalent” (Moishezon 1994, p. 155). Hence, if one found an infinite number of non-equivalent factorizations of braids, when each element is (possibly a conjugation of) a braid of exponent 1, 2 or 3, and when every factorization contains the same number of elements with exponents 1, 2 and 3, then “infinitely many of [the factorizations] are not analytic (that is, do not correspond to braid monodromies of cuspidal curves)” (ibid., p. 156). The key step in Moishezon’s proof of the theorem was to prove the non-equivalence of the infinite number of factorizations he found, using methods from group theory.

  45. I follow here De Toffoli and Giardino, who claim (regarding knot diagrams) that “[t]he use of diagrams triggers a form of manipulative imagination that gets enhanced by the practice” (De Toffoli and Giardino 2014, p. 836; see also Giardino 2018). However, my point is to note that what might be “enhanced by the practice” could lead to wrong results or to a dead end.

  46. See also Schappacher (2008, 2011, pp. 3276–3286) for a more general account of this rewriting process.

  47. The notation of Tait (1876) might be thought as another case study of a notation of braids and their crossings; Tait’s notation, being global, was similar in a way to Dedò’s. Moreover, Tait did suggest thinking about braids, which turn in reverse, though with the notation of these braids, “the application of this method becomes very troublesome” (ibid., p. 243). Though Tait did formulate few relations between the braids, his research, however, culminated in a dead end, since he did not consider the set of braids as having an algebraic structure. See Epple (1999, pp. 139–140).

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Acknowledgements

A preliminary version of this paper was presented at the Symposium “Mathematics and Practice. Historical, philosophical and educational perspectives” at the University of Rostock, March 22–23, 2019. I thank warmly all the participants for their helpful comments. The research for this paper was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—EXC 2015/1.

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  1. Cluster of Excellence “Matters of Activity. Image Space Material” at the Humboldt-Universität zu Berlin, Sophienstr. 22a, 10178, Berlin, Germany

    Michael Friedman

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Friedman, M. How to notate a crossing of strings? On Modesto Dedò’s notation of braids. Arch. Hist. Exact Sci. 74, 281–312 (2020). https://doi.org/10.1007/s00407-019-00238-8

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