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Virtual strings

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  • Mathematics

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Virtual strings AN N A L E S D E L’INSTI T U T F O U R IE R ANNALES DE L’INSTITUT FOURIER Vladimir TURAEV Virtual strings Tome 54, no 7 (2004), p. 2455-2525. <http://aif.cedram.org/item?id=AIF_2004__54_7_2455_0> © Association des Annales de l’institut Fourier, 2004, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ 2455 VIRTUAL STRINGS by Vladimir TURAEV Contents. 1. Introduction. A virtual string is a scheme of self-intersections of a generic oriented closed curve on an oriented surface. More precisely, a virtual string of rank m &#x3E; 0 is an oriented circle with 2m distinguished points partitioned into m ordered pairs. These m ordered pairs of points are called arrows of the virtual string. An example of a virtual string of rank 3 is shown on Figure 1 where the arrows are represented by geometric vectors. A (generic oriented) closed curve on an oriented surface gives rise to an "underlying" virtual string whose arrows correspond to the self-crossings Keywords: Virtual strings - Virtual knots - Surfaces - Cobordism - Skew-symmetric matrices - Lie cobracket. Math. classification: 57M99. 2456 Figure 1. A virtual string of rank 3 of the curve. The usual homotopy of curves on surfaces suggests a notion of homotopy for strings. The homotopy of curves in 3-manifolds with boundary suggests a notion of cobordism for strings. The main objective of the theory of

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