Japanese Journal of Mathematics

, Volume 2, Issue 1, pp 1–39 | Cite as

Lectures on topology of words

original article


We discuss a topological approach to words introduced by the author in [Tu2]–[Tu4]. Words on an arbitrary alphabet are approximated by Gauss words and then studied up to natural modifications inspired by the Reidemeister moves on knot diagrams. This leads us to a notion of homotopy for words. We introduce several homotopy invariants of words and give a homotopy classification of words of length five.

Keywords and phrases:

words curves homotopy 

Mathematics Subject Classification (2000):

57M99 68R15 


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  1. G. Cairns and D. M. Elton, The planarity problem for signed Gauss words, J. Knot Theory Ramifications, 2 (1993), 359–367.MATHCrossRefMathSciNetGoogle Scholar
  2. J. S. Carter, S. Kamada and M. Saito, Stable equivalence of knots on surfaces and virtual knot cobordisms, J. Knot Theory Ramifications, 11 (2002), 311–322.MATHCrossRefMathSciNetGoogle Scholar
  3. N. Chaves and C. Weber, Plombages de rubans et probléme des mots de Gauss, Expo. Math., 12 (1994), 53–77 and 124.MATHMathSciNetGoogle Scholar
  4. H. Crapo and P. Rosenstiehl, On lacets and their manifolds, Discrete Math., 233 (2001), 299–320.MATHCrossRefMathSciNetGoogle Scholar
  5. C. H. Dowker and M. B. Thistlethwaite, Classification of knot projections, Topology Appl., 16 (1983), 19–31.MATHCrossRefMathSciNetGoogle Scholar
  6. C. F. Gauss, Werke, Vol. VIII, Teubner, Leipzig, 1900, pp. 272, 282–286.Google Scholar
  7. M. Goussarov, M. Polyak and O. Viro, Finite-type invariants of classical and virtual knots, Topology, 39 (2000), 1045–1068.MATHCrossRefMathSciNetGoogle Scholar
  8. S. Kamada, Knot invariants derived from quandles and racks. Invariants of knots and 3-manifolds (Kyoto, 2001), 103–117 (electronic), Geom. Topol. Monogr., 4, Geom. Topol. Publ., Coventry, 2002.Google Scholar
  9. N. Kamada and S. Kamada, Abstract link diagrams and virtual knots, J. Knot Theory Ramifications, 9 (2000), 93–106.MATHCrossRefMathSciNetGoogle Scholar
  10. L. Kauffman, Virtual knot theory, European J. Combin., 20 (1999), 663–690.MATHCrossRefMathSciNetGoogle Scholar
  11. L. Lovász and M. L. Marx, A forbidden substructure characterization of Gauss codes, Acta Sci. Math. (Szeged), 38 (1976), 115–119.MATHMathSciNetGoogle Scholar
  12. M. L. Marx, The Gauss realizability problem, Proc. Amer. Math. Soc., 22 (1969), 610–613.MATHCrossRefMathSciNetGoogle Scholar
  13. P. Rosenstiehl, Solution algébrique du problème de Gauss sur la permutation des points d’intersection d’une ou plusieurs courbes fermées du plan, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), A551–A553.MathSciNetGoogle Scholar
  14. J. Sawollek, On Alexander-Conway polynomials for virtual knots and links, math.GT/9912173.Google Scholar
  15. D. Silver and S. Williams, Polynomial invariants of virtual links, J. Knot Theory Ramifications, 12 (2003), 987–1000.MATHCrossRefMathSciNetGoogle Scholar
  16. D. Silver and S. Williams, An invariant for open virtual strings, J. Knot Theory Ramifications, 15 (2006), 143–152.MATHCrossRefMathSciNetGoogle Scholar
  17. V. Turaev, Virtual strings, Ann. Inst. Fourier, 54 (2004), 2455–2525.MATHMathSciNetGoogle Scholar
  18. V. Turaev, Topology of words, math.CO/0503683.Google Scholar
  19. V. Turaev, Knots and words, math.GT/0506390.Google Scholar
  20. V. Turaev, Cobordism of words, math.CO/0511513.Google Scholar

Copyright information

© The Mathematical Society of Japan and Springer 2007

Authors and Affiliations

  1. 1.IRMAUniversité Louis Pasteur - C.N.R.S.StrasbourgFrance

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