Skip to main content
SpringerLink
Log in

Braids and Knots

  • Chapter
  • First Online:
Lectures on Topological Fluid Mechanics

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1973))

  • 2247 Accesses

Abstract

We introduce braids via their historical roots and uses, make connections with knot theory and present the mathematical theory of braids through the braid group. Several basic mathematical properties of braids are explored and equivalence problems under several conditions defined and partly solved. The connection with knots is spelled out in detail and translation methods are presented. Finally a number of applications of braid theory are given. The presentation is pedagogical and principally aimed at interested readers from different fields of mathematics and natural science. The discussions are as self-contained as can be expected within the space limits and require very little previous mathematical knowledge. Literature references are given throughout to the original papers and to overview sources where more can be learned.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adams, C., Hildebrand, M., Weeks, J. (1991): Hyperbolic Invariants of Knots and Links. Trans. Amer. Math. Soc., 326, 1–56.

    Article  MATH  MathSciNet  Google Scholar 

  2. Adian, S. I. (1957): The unsolvability of certain algorithmic problems in the theory of groups. Trudy Moskov. Mat. Obsc., 6, 231–298.

    MathSciNet  Google Scholar 

  3. Adian, S. I. (1957): Finitely Presented Group and Algorithms. Dokl. Akad. Nauk SSSR, 117, 9–12.

    MathSciNet  Google Scholar 

  4. Alexander, J. W. (1923): A lemma on systems of knotted curves. Proc. Nat. Acad. Sci. USA, 9, 93–95.

    Article  ADS  Google Scholar 

  5. Appel, K. I. and Schupp, P. E. (1972): The Conjugacy Problem for the Group of any Tame Alternating Knot is Solvable. Proc. Amer. Math. Soc., 33, 329–336.

    Article  MATH  MathSciNet  Google Scholar 

  6. Artin, E. (1925): Theorie der Zöpfe. (in German) Abh. Math. Sem. Univ. Hamburg, 4, 47–72.

    Article  MATH  Google Scholar 

  7. Artin, E. (1947): Theory of braids. Ann. Math., 48, 101–126.

    Article  MATH  MathSciNet  Google Scholar 

  8. Baader, F., Nipkow, T. (1998): Term Rewriting and All That. (Cambridge University Press, Cambridge).

    Google Scholar 

  9. Bangert, P. D., Berger, M. A., Prandi, R. (2002): In Search of Minimal Random Braid Configurations. J. Phys. A., 35, 43–59.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Berger, M. A. (1993): Energy-crossing number relations for braided magnetic fields. Phys. Rev. Lett., 70, 705–708.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Berger, M. A. (1994): Minimum crossing numbers for 3-braids. J. Phys. A, 27, 6205–6213.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. Berger, M.A. (2000): Hamiltonian dynamics generated by Vassiliev invariants. J. Phys. A., 34, 1363–1374.

    Article  ADS  Google Scholar 

  13. Bigelow, S. (1999): The Burau representation is not faithful for n = 5. Geometry and Topology, 3, 397–404.

    Article  MATH  MathSciNet  Google Scholar 

  14. Birman, J. S. (1974): Braids, Links and Mapping Class Groups. Ann. of Math. Studies 82 (Princeton Univ. Press, Princeton).

    Google Scholar 

  15. Birman, J. S., Ko, K. H., Lee, S. J. (1998): A New Approach to the Word and Conjugacy Problems in the Braid Groups. Ad. Math., 139, 322–353.

    Article  MATH  MathSciNet  Google Scholar 

  16. Birman, J. S., Menasco, W. (1992): Studying Links Via Closed Braids I: A Finiteness Theorem. Pacific J. Math., 154, 17–36.

    MATH  MathSciNet  Google Scholar 

  17. Birman, J. S., Menasco, W. (1991): Studying Links Via Closed Braids II: On a Theorem of Bennequin. Topology and Its Applications, 40, 71–82.

    Article  MATH  MathSciNet  Google Scholar 

  18. Birman, J. S., Menasco, W. (1993): Studying Links Via Closed Braids III: Classifying Links which are Closed 3-braids. Pacific J. Math., 161, 25–113.

    MATH  MathSciNet  Google Scholar 

  19. Birman, J. S., Menasco, W. (1990): Studying Links Via Closed Braids IV: Closed Braid Representatives of Split and Composite Links. Invent. Math., 102, 115–139.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. Birman, J. S., Menasco, W. (1992): Studying Links Via Closed Braids V: Closed Braid Representatives of the Unlink. Trans. AMS, 329, 585–606.

    MATH  MathSciNet  Google Scholar 

  21. Birman, J. S., Menasco, W. (1992): Studying Links Via Closed Braids VI: A Non-Finiteness Theorem. Pacific J. Math., 156, 265–285.

    MATH  MathSciNet  Google Scholar 

  22. Birman, J. S., Menasco, W. (1992): A calculus on links in the 3-sphere. in Kawauchi, A. Knots 90 (Walter de Gruyter, Berlin), 625–631.

    Google Scholar 

  23. Birman, J. S., Wajnryb, B. (1986): Markov Classes in Certain Finite Quotients of Artin's Braid Group. Israel J. Math., 56, 160–178.

    Article  MATH  MathSciNet  Google Scholar 

  24. Birkhoff, G. (1935): On the structure of abstract algebras. Proc. Camb. Phil. Soc., 31, 433–454.

    Article  ADS  Google Scholar 

  25. Bohnenblust, F. (1947): The Algebraical Braid Group. Ann. Math., 46, 127–136

    Article  MathSciNet  Google Scholar 

  26. Boyland, P. L., Aref, H., Stremler, M. A. (2000): Topological fluid mechanics of stirring. J. Fluid Mech., 403, 277–304.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  27. Buchberger, B. (1987): History and Basic Features of the Critical-Pair/Completion Procedure. J. Sym. Comp. 3, 3–38. Printed in Rewriting Techniques and Applications ed. by Jouannaud, J.-P. (Academic Press, London), 3–38.

    Article  MATH  MathSciNet  Google Scholar 

  28. Chan, T. (2000): HOMFLY polynomials of some generalized Hopf links. J. Knot Th. Rami., 9, 865–883.

    Article  MATH  Google Scholar 

  29. Cohen, D.E. (1987): Computability and Logic. (Ellis Horwood, Chichester).

    MATH  Google Scholar 

  30. Conway, J.H. (1970): An Enumeration of Knot and Links, and Some of their Algebraic Properties. in Leech, J. (1970): Computational Problems in Abstract Algebra. (Pergamon, Oxford)

    Google Scholar 

  31. Coxeter, H.S.M., Moser, W.O.J. (1957): Generators and Relations for Discrete Groups. (Springer, Berlin)

    MATH  Google Scholar 

  32. Dershowitz, N. (1979): A note on simplification orderings. Inform. Proc. Let., 9, 212–215.

    Article  MATH  MathSciNet  Google Scholar 

  33. Dershowitz, N. (1981): Termination of linear rewriting systems. in Automata, Languages and Programming ed. by Even, S. and Kariv, O., Lecture Notes in Computer Science Volume 115 (Springer, Heidelberg), 448–458.

    Chapter  Google Scholar 

  34. Dershowitz, N. (1987): Termination of Rewriting. J. Sym. Comp., 3, 69–116. Printed in Rewriting Techniques and Applications ed. by Jouannaud, J.-P. (Academic Press, London), 69–116.

    Article  MATH  MathSciNet  Google Scholar 

  35. Dowker, C. H. and Thistlethwaite, M. B. (1983): Classification of Knot Projections. Top. Appl., 16, 19–31.

    Article  MATH  MathSciNet  Google Scholar 

  36. Epstein, D.B.A., Cannon, J.W., Holt, D.F., Levy, S.V.F., Paterson, M.S., Thurston, W.P. (1992): Word Processing in Groups. (Jones and Bartlett, Boston).

    MATH  Google Scholar 

  37. Garside, F.A. (1969): The braid group and other groups. Quart. J. Math. Oxford, 20, 235–254.

    Article  MATH  MathSciNet  Google Scholar 

  38. Gilbert, N.D., Porter, T. (1994): Knots and Surfaces. (Oxford University Press, Oxford).

    Google Scholar 

  39. Hemion, G. (1992): The Classification of Knots and 3-dimensional Spaces. (Oxford Univ. Press, Oxford).

    MATH  Google Scholar 

  40. Hempel, J. (1976): 3-manifolds. Ann. of Math. Studies Volume 86 (Princeton Uni. Press, Princeton).

    Google Scholar 

  41. Huet, G. (1980): Confluent reductions: Abstract properties and applications to term rewriting systems. J. Assoc. Comput. Mach., 27, 797–821.

    Article  MATH  MathSciNet  Google Scholar 

  42. Huet, G. (1981): A Complete Proof of the Knuth-Bendix Completion Algorithm. J. Comp. Syst. Sci., 23, 11–21.

    Article  MATH  MathSciNet  Google Scholar 

  43. Huet, G., Lankford, D.S. (1978): On the uniform halting problem for term rewriting systems. Rapport laboria 283, Institut de Recherche en Informatique et en Automatique, Le Chesnay, France.

    Google Scholar 

  44. Jacquemard, A. (1990): About the effective classification of conjugacy classes of braids. J. Pure Appl. Al., 63, 161–169.

    Article  MATH  MathSciNet  Google Scholar 

  45. Johnson, D.L. (1980): Topics in the Theory of Group Presentations. Lon. Math. Soc. Lec. Notes Vol. 42 (Cambridge Uni. Press, Cambridge).

    Book  MATH  Google Scholar 

  46. Jouannoud, J.P., Kirchner, H. (1986): Completion of a set of rules modulo a set of equations, SIAM. J. Comp., 15, 1155–1194.

    Google Scholar 

  47. Kang, E. S. et al. (1997): Band-generator presentation for the 4-braid group. Top. Appl., 78, 39–60.

    Article  MATH  Google Scholar 

  48. Kauffman, L. (1993): Knots and Physics. Series on Knots and Everything Vol. 1. World Scientific, Singapore.

    Google Scholar 

  49. Kawauchi, A. (1996): A Survey of Knot Theory. (Birkhäuser Verlag, Basel)

    MATH  Google Scholar 

  50. Knuth, D.E., Bendix, P.B. (1970): Simple word problems in universal algebras. in Computational Problems in Abstract Algebra ed. by Leech, J. (Pergamon Press, Oxford), 263–297. Reprinted in 1983 in Automation of Reasoning 2 (Springer, Berlin), 342–376.

    Google Scholar 

  51. Lambropoulou, S.S.F. (1993): A Study of Braid in 3-manifolds. unpublished PhD thesis (Univ. of Warwick).

    Google Scholar 

  52. Lambropoulou, S.S.F., Rourke, C.P. (1997): Markov's Theorem in 3-manifolds. Top. Appl., 78, 95–22.

    Article  MATH  MathSciNet  Google Scholar 

  53. The CiME system is available from http://cime.lri.fr.

  54. Magnus, W., Peluso, A. (1967): On Knot Groups. Comm. Pure Appl. Math., 20, 749–770.

    Article  MATH  MathSciNet  Google Scholar 

  55. Markov, A.A. (1935): Über die freie Äquivalenz geschlossener Zöpfe (in German), Recueil Mathematique Moscou, 1, 73–78. [Mat. Sb. 43 1936.]

    Google Scholar 

  56. McCool, J. (1980): On Reducible Braids. in Word Problems II ed. by Adian, S. I., Boone, W. W. and Higman, G. (North-Holland, Amsterdam), 261–295.

    Chapter  Google Scholar 

  57. Miller, C.F. III (1992): Decision Problems for Groups - Survey and Reflections. in Algorithms and Classification in Combinatorial Group Theory ed. by Baumslag, G. and Miller, C. F. III, Math. Sci. Re. Inst. Pub. Volume 23 (Springer, New York), 1–59.

    Chapter  Google Scholar 

  58. Morton, H.R. (1983): An irreducible 4-string braid with unknotted closure. Math. Proc. Camb. Phil. Soc., 93, 259–261.

    Article  MATH  MathSciNet  Google Scholar 

  59. Morton, H.R. (1986): Threading Knot Diagrams. Math. Proc. Camb. Phil. Soc., 99, 247–260.

    Article  MATH  MathSciNet  Google Scholar 

  60. Murasugi, K. (1996): Knot Theory And Its Applications. (Birkhäuser, Boston).

    MATH  Google Scholar 

  61. Murasugi, K., Thomas, R.S.D. (1972): Isotopic closed nonconjugate braids. Proc. Am. Math. Soc., 33, 137–139.

    Article  MATH  MathSciNet  Google Scholar 

  62. Newman, M.H.A. (1942): On theories with a combinatorial definition of ‘equivalence’. Ann. Math., 43, 223–243.

    Article  MATH  Google Scholar 

  63. Paterson, M.S., Razborov, A.A. (1991): The set of minimal braids is co-NP-complete. J. Algorithms, 12, 393–408.

    Article  MATH  MathSciNet  Google Scholar 

  64. Rabin, M.O. (1958): Recursive Unsolvability of Group Theoretic Problems. Ann. Math., 67, 172–194.

    Article  MATH  MathSciNet  Google Scholar 

  65. Reidemeister, K. (1983): Knot Theory. BCS Associates, Moscow, Idaho. Originally published as Reidemeister, K. (1932): Knotentheorie. Springer, Berlin

    MATH  Google Scholar 

  66. Ricca, R.L. (1998): Applications of Knot Theory in Fluid Mechanics. in Jones, V.F.R. et. al., ed. by Knot Theory Banach Center Pub. Vol. 42 (Inst. of Math., Polish Acad. Sci., Warszawa), 321–346.

    Google Scholar 

  67. Schubert, H. (1948): Die Eindeutige Zerlegbarkeit eines Knotens in Primknoten. (in German), Sitz. Heidelberger Akad. Wiss., math.-nat. Kl., 55–104.

    Google Scholar 

  68. Schubert, H. (1961): Bestimmung der Primfaktorzerlegung von Verkettungen. (in German), Math. Zeitschr., 76, 116–148.

    Article  MATH  MathSciNet  Google Scholar 

  69. Tatsuoka, K. (1987): Geodesics in the braid group. preprint, Dept. of Mathematics, University of Texas at Austin.

    Google Scholar 

  70. Tourlakis, G.J. (1984): Computability. (Reston, Reston).

    MATH  Google Scholar 

  71. Turing, A.M. (1937): On Computable Numbers, with an Application to the Entscheidungsproblem. Proc. London Math. Soc. Ser. 2, 42, 230–265.

    Article  Google Scholar 

  72. Vogel, P. (1990): Representation of links by braids: A new algorithm. Comment. Math. Helvetici, 65, 104–113.

    Article  MATH  Google Scholar 

  73. Waldhausen, F. (1968): On Irreducible 3-manifolds which are Sufficiently Large. Ann. Math., 87, 56–88.

    Article  MATH  MathSciNet  Google Scholar 

  74. Waldhausen, F. (1968): The Word Problem in Fundamental Groups of Sufficiently Large Irreducible 3-manifolds. Ann. Math., 88, 272–280.

    Article  MATH  MathSciNet  Google Scholar 

  75. Williams, R.F. (1988): The Braid Index of an Algebraic Link. In: Birman, J. S., Libgober, A. (ed) Braids. Amer. Math. Soc., Providence.

    Google Scholar 

  76. Xu, P.J. (1992): The genus of closed 3-braids. J. Knot Theory Ramifications, 1, 303–326.

    Article  MATH  MathSciNet  Google Scholar 

  77. Yamada, S. (1987): The minimal number of Seifert circles equals the braid index of a link. Invent. Math., 89, 347–356.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  78. Yoder, M.A. (1995): String Rewriting Applied to Problems in the Braid Groups. unpublished Ph.D. thesis (Uni. South Florida).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisch Instituut, Ausser der Schleifmühle 67, 28203, Germany

    Patrick D. Bangert

Authors
  1. Patrick D. Bangert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Patrick D. Bangert .

Editor information

Editors and Affiliations

  1. Dipto. Matematica e Applicazioni, Università Milano-Bicocca, Via R. Cozzi 53, Milano, 20125, Italy

    Renzo L. Ricca

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Bangert, P.D. (2009). Braids and Knots. In: Ricca, R. (eds) Lectures on Topological Fluid Mechanics. Lecture Notes in Mathematics(), vol 1973. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00837-5_1

Download citation

Publish with us

Policies and ethics

Search

Navigation