no. 7
Virtual strings
[Cordes virtuelles]
Turaev, Vladimir1
1 Université Louis Pasteur, IRMA, CNRS, 7 rue René Descartes, 67084 Strasbourg (France)
Annales de l'Institut Fourier, Tome 54 (2004) no. 7, pp. 2455-2525.

Une corde virtuelle est un schéma d'auto-intersections d'une courbe fermée sur une surface. Nous étudions les invariants algébriques des cordes virtuelles ainsi que deux relations d'équivalence sur l'ensemble des cordes : homotopie et cobordisme. Nous montrons que les invariants d'homotopie de cordes forment un groupe de Lie de dimension infinie. Les rapports avec les noeuds virtuels sont également abordés.

A virtual string is a scheme of self-intersections of a closed curve on a surface. We study algebraic invariants of strings as well as two equivalence relations on the set of strings: homotopy and cobordism. We show that the homotopy invariants of strings form an infinite dimensional Lie group. We also discuss connections between virtual strings and virtual knots.

DOI : 10.5802/aif.2086
Classification : 57M99
Keywords: Virtual strings, virtual knots, surfaces, cobordism, skew-symmetric matrices, Lie cobracket
Mot clés : cordes virtuelles, noeuds virtuels, surfaces, cobordisme, matrices antisymétriques, co-crochet de Lie
Turaev, Vladimir 1

1 Université Louis Pasteur, IRMA, CNRS, 7 rue René Descartes, 67084 Strasbourg (France) 
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Turaev, Vladimir. Virtual strings. Annales de l'Institut Fourier, Tome 54 (2004) no. 7, pp. 2455-2525. doi : 10.5802/aif.2086. http://www.numdam.org/articles/10.5802/aif.2086/

[Ca1] J.S. Carter Classifying immersed curves, Proc. Amer. Math. Soc., Volume 111 (1991) no. 1, pp. 281-287 | DOI | MR | Zbl

[Ca2] J.S. Carter Closed curves that never extend to proper maps of disks, Proc. Amer. Math. Soc., Volume 113 (1991) no. 3, pp. 879-888 | DOI | MR | Zbl

[CR] P. Cotta-Ramusino; M. Rinaldi On the algebraic structure of link-diagrams on a 2-dimensional surface, Comm. Math. Phys., Volume 138 (1991) no. 1, pp. 137-173 | DOI | MR | Zbl

[CSK] J.S. Carter; S. Kamada; M. Saito Stable equivalence of knots on surfaces and virtual knot cobordisms, J. Knot Theory Ramifications, Volume 11 (2002) no. 3, pp. 311-322 | DOI | MR | Zbl

[CW] N. Chaves; C. Weber Plombages de rubans et problème des mots de Gauss, Exposition. Math., Volume 12 (1994) no. 1, pp. 53-77 | MR | Zbl

[CW] N. Chaves; C. Weber Erratum "Plombages de rubans et problème des mots de Gauss", Exposition. Math., Volume 12 (1994) no. 2, pp. 124 | MR | Zbl

[Fr] J.K. Francis The folded ribbon theorem. A contribution to the study of immersed circles, Trans. Amer. Math. Soc., Volume 141 (1969), pp. 271-303 | DOI | MR | Zbl

[GPV] M. Goussarov; M. Polyak; O. Viro Finite-type invariants of classical and virtual knots, Topology, Volume 39 (2000) no. 5, pp. 1045-1068 | DOI | MR | Zbl

[HK] D. Hrencecin; L. Kauffman On Filamentations and Virtual Knots, Topology and Its Applications, Volume 134 (2003), pp. 23-52 | DOI | MR | Zbl

[Ja] F. Jaeger Letter to the author (21-11-1991)

[Ka] L. Kauffman Virtual knots theory, European J. Combin., Volume 20 (1999) no. 7, pp. 663-690 | DOI | MR | Zbl

[Pr] J. Przytycki Quantum group of links in a handlebody, Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990) (Contemp. Math.), Volume 134 (1992), pp. 235-245 | Zbl

[Se] J.-P. Serre Lie algebras and Lie groups, Lecture Notes in Math., 1500, Springer-Verlag, Berlin, 1992 | MR | Zbl

[Tu1] V. Turaev The Yang-Baxter equation and invariants of links, Invent. Math., Volume 92 (1988) no. 3, pp. 527-553 | DOI | MR | Zbl

[Tu2] V. Turaev Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. École Norm. Sup. (4), Volume 24 (1991) no. 6, pp. 635-704 | Numdam | MR | Zbl

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