Virtual strings

Vladimir Turaev[1]

  • [1] Université Louis Pasteur, IRMA, CNRS, 7 rue René Descartes, 67084 Strasbourg (France)

Annales de l'Institut Fourier (2004)

  • Volume: 54, Issue: 7, page 2455-2525
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Turaev, Vladimir. "Virtual strings." Annales de l'Institut Fourier 54.7 (2004): 2455-2525. <http://eudml.org/doc/116179>.

@article{Turaev2004,
abstract = {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.},
affiliation = {Université Louis Pasteur, IRMA, CNRS, 7 rue René Descartes, 67084 Strasbourg (France)},
author = {Turaev, Vladimir},
journal = {Annales de l'Institut Fourier},
keywords = {Virtual strings; virtual knots; surfaces; cobordism; skew-symmetric matrices; Lie cobracket; virtual strings; homotopy},
language = {eng},
number = {7},
pages = {2455-2525},
publisher = {Association des Annales de l'Institut Fourier},
title = {Virtual strings},
url = {http://eudml.org/doc/116179},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Turaev, Vladimir
TI - Virtual strings
JO - Annales de l'Institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 7
SP - 2455
EP - 2525
AB - 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.
LA - eng
KW - Virtual strings; virtual knots; surfaces; cobordism; skew-symmetric matrices; Lie cobracket; virtual strings; homotopy
UR - http://eudml.org/doc/116179
ER -

References

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  8. D. Hrencecin, L. Kauffman, On Filamentations and Virtual Knots, Topology and Its Applications 134 (2003), 23-52 Zbl1031.57008MR2005847
  9. F. Jaeger, Letter to the author, (21-11-1991) 
  10. L. Kauffman, Virtual knots theory, European J. Combin. 20 (1999), 663-690 Zbl0938.57006MR1721925
  11. J. Przytycki, Quantum group of links in a handlebody, Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990) 134 (1992), 235-245, Amer. Math. Soc., Providence, RI Zbl0779.57004
  12. J.-P. Serre, Lie algebras and Lie groups, 1500 (1992), Springer-Verlag, Berlin Zbl0742.17008MR1176100
  13. V. Turaev, The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988), 527-553 Zbl0648.57003MR939474
  14. V. Turaev, Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. École Norm. Sup. (4) 24 (1991), 635-704 Zbl0758.57011MR1142906
  15. N. Chaves, C. Weber, Erratum "Plombages de rubans et problème des mots de Gauss", Exposition. Math. 12 (1994) Zbl0816.57011MR1267628

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