Skip to main content

Combinatorial Recoupling Theory and 3-Manifold Invariants

  • Chapter
Low-Dimensional Topology and Quantum Field Theory

Part of the book series: NATO ASI Series ((NSSB,volume 315))

Abstract

This paper discusses combinatorial recoupling theory, first in relation to the vector cross product algebra and a reformulation of the Four Colour Theorem, and secondly in relation to the Temperley-Lieb algebra, the Jones polynomial and the SU(2) 3-Manifold invariants of Witten, Reshetikhin and Turaev.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover 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.

Similar content being viewed by others

References

  1. L.C. Biedenharn and J.D. Louck, Angular momentum in quantum physics — theory and application, in “Encyclopaedia of Mathematics and its Applications”, Cambridge University Press (1979).

    Google Scholar 

  2. L. Crane, Conformal Field Theory, Spin Geometry and Quantum Gravity, Phys. Lett. B 259 (1991) 243–248.

    Article  MathSciNet  ADS  Google Scholar 

  3. R.A. Fenn and C.P. Rourke, On Kirby’s Calculus of Links, Topology 18 (1979) 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  4. B. Hasslacher and M.J. Perry, Spin Networks are Simplicial Quantum Gravity, Phys. Lett. B 103 (1981) 21–24.

    Article  MathSciNet  ADS  Google Scholar 

  5. V.F.R. Jones, Index for subfactors, Invent. Math. 72 (1983) 1–25.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. V.F.R. Jones, A polynomial invariant for links via von Neumann algebras, Bull. Amer. Math. Soc. 129 (1985) 103–112.

    Article  Google Scholar 

  7. V.F.R. Jones, A New Knot Polynomial and von Neumann algebras, Notices of AMS 33 (1986) 219–225.

    Google Scholar 

  8. V.F.R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (1987) 335–38.

    Article  MathSciNet  MATH  Google Scholar 

  9. L.H. Kauffman, State Models and the Jones Polynomial, Topology 26 (1987) 395–407.

    Article  MathSciNet  MATH  Google Scholar 

  10. L.H. Kauffman, On Knots, Annals of Mathematics Studies Number 115, Princeton University Press (1987).

    Google Scholar 

  11. L.H. Kauffman, New invariants in the theory of knots, Amer. Math. Monthly 95 (1988) 195–242.

    Article  MathSciNet  MATH  Google Scholar 

  12. L.H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990) 417–471.

    Article  MathSciNet  MATH  Google Scholar 

  13. L.H. Kauffman, Spin Networks and Knot Polynomials, Int. J. Mod. Phys. A 5 (1990) 93–115.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. L.H. Kauffman, Map Colouring and the Vector Cross Product, J. Comb. Theo. Ser. B. 48 (1990) 145–154.

    Article  MathSciNet  MATH  Google Scholar 

  15. L.H. Kauffman, Knots and Physics, World Scientific (1991).

    Google Scholar 

  16. L.H. Kauffman, Knots, Spin Networks and 3-Manifold Invariants, Knots 90 (edited by A. Kawauchi) (1992) pp. 271-287.

    Google Scholar 

  17. L.H. Kauffman, Map Colouring, q-deformed Spin Networks, and Turaev-Viro Invariants for 3-manifolds, In The Proceedings of the Conference on Quantum Groups Como, Italy, June 1991, Ed. M. Rasetti — World Sci. Pub., Int. J. Mod. Phys. B 6 (1992) 1765–1794.

    Google Scholar 

  18. L.H. Kauffman and S. Lins, A 3-manifold invariant by state summation, (announcement 1990).

    Google Scholar 

  19. L.H. Kauffman and S. Lins, Computing Turaev-Viro Invariants for 3-Manifolds. Manuscripta Math. 72 (1991) 81–94.

    Article  MathSciNet  MATH  Google Scholar 

  20. L.H. Kauffman and S. Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, (preprint 1993).

    Google Scholar 

  21. L.H. Kauffman and H. Saleur, Map colouring and the Temperley-Lieb algebra, (To appear in Comm. Math. Phys).

    Google Scholar 

  22. R. Kirby, A calculus for framed links in S 3, Invent. Math. 45 (1978) 35–56.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. R. Kirby and P. Melvin, On the 3-manifold invariants of Reshetikhin-Turaev for sl(2, C), Invent. Math. 105 (1991) 473–545

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. A.N. Kirillov and N.Y. Reshetikhin, Representations of the algebra U q (s 12), q-orthogonal polynomials and invariants of links. In Infinite Dimensional Lie Algebras and Groups ed. by V.G. Kac. Adv. Ser. in Math. Phys. 7 (1988) 285.

    Google Scholar 

  25. S.I. Kryuchkov, The four-colour theorem and trees, I.V. Kurchatov Institute of Atomic Energy, Moscow (1992) — IAE — 5537/1.

    Google Scholar 

  26. W.B.R. Lickorish, A Representation of Orientable Combinatorial 3-Manifolds, Ann. Maths. 76 (1992) 51.

    MathSciNet  Google Scholar 

  27. W.B.R. Lickorish, Calculations with the Temperley-Lieb algebra, Commetarii Mathematici Helvetia 67 (1992) 571.

    Article  MathSciNet  MATH  Google Scholar 

  28. W.B.R. Lickorish, 3-Manifolds and the Temperley-Lieb Algebra. Math. Ann. 290 (1991) 657.

    Article  MathSciNet  MATH  Google Scholar 

  29. W.B.R. Lickorish, Skeins and handlebodies, (preprint 1992).

    Google Scholar 

  30. W.B.R. Lickorish, The Temperley-Lieb Algebra and 3-manifold invariants, (preprint 1992), (and private communication).

    Google Scholar 

  31. S. Lins and C. Durand, Topological classification of small graph-encoded orientable 3-manifolds, Notas Comm. Mat. UFPE, 177 (1991).

    Google Scholar 

  32. G. Masbaum and P. Vogel, 3-valent graphs and the Kauffman bracket, (preprint 1992).

    Google Scholar 

  33. J.P. Moussouris, The Chromatic Evaluation of Strand Networks. Advances in Twistor Theory by Huston and Ward Research Notes in Mathematics, Pitman Pub. (1979), pp. 308-312.

    Google Scholar 

  34. J.P. Moussouris, Quantum models of space-time based on recoupling theory, (Mathematics Thesis, Oxford University — 1983).

    Google Scholar 

  35. H. Ooguri and N. Sasakura, Discrete and Continuum approaches to three-dimensional quantum gravity, Mod. Phys. Lett. A6 (1991) 3591.

    MathSciNet  ADS  Google Scholar 

  36. R. Penrose, Angular momentum: an approach to combinatorial space-time, in “Quantum Theory and Beyond”, T.A. Bastin ed., Cambridge Univ. Press (1969).

    Google Scholar 

  37. R. Penrose, Applications of negative dimensional tensors, in “Combinatorial Mathematics and its Applications”, D.J.A. Welsh ed., Academic Press (1971).

    Google Scholar 

  38. R. Penrose, Combinatorial quantum theory and quantized directions, in “Advances in Twistor Theory”, L.P. Hughston and R.S. Ward eds., Pitman (1979), pp. 301-307.

    Google Scholar 

  39. Sergey Piunikhin, Turaev-Viro and Kauffman-Lins invariants for 3-manifolds coincide, Journal of Knot Theory and its Ramifications, 1 (1992) 105.

    Article  MathSciNet  MATH  Google Scholar 

  40. N.Y. Reshetikhin, Quantized universal enveloping algebras, the Yang-Baxter equation and the invariants of links, 1 and 11, LOMI reprints E-4-87 and E-17-87, Steklov Institute, Leningrad, USSR.

    Google Scholar 

  41. N.Y. Reshetikhin and V. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990) 1.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. N.Y. Reshetikhin and V. Turaev, Invariants of three manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991) 547.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  43. V.G. Turaev, The Yang-Baxter equations and invariants of links, LOMI preprint E-3-87, Steklov Institute, Leningrad, USSR, Inventiones Math. 92 Fasc. 3, 527.

    Google Scholar 

  44. V.G. Turaev and O. Viro, State sum invariants of 3-manifolds and quantum 6j symbols, Topology 31 1992 865.

    Google Scholar 

  45. V.G. Turaev Quantum invariants of links and 3-valent graphs in 3-manifolds, (preprint 1990).

    Google Scholar 

  46. V.G. Turaev, Quantum invariants of 3-manifolds and a glimpse of shadow topology, Comptes Rendus de l’Academic des Sciences, Serie I-Math. 313 (1991) 395.

    MathSciNet  MATH  Google Scholar 

  47. V.G. Turaev, Shadow links and face models of statistical mechanics, J. Diff. Geom. 36 (1992) 35.

    MathSciNet  MATH  Google Scholar 

  48. V.G. Turaev, Topology of shadows, (preprint 1992).

    Google Scholar 

  49. K. Walker, On Witten’s 3-manifold invariants, (preprint 1991).

    Google Scholar 

  50. R. Williams and F. Archer, The Turaev-Viro state sum model and 3-dimensional quantum gravity, Phys. Lett. B 273 (1991) 438.

    Article  MathSciNet  ADS  Google Scholar 

  51. E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  52. S. Yamada, A Topological invariant of spatial regular graphs, in “Knots’ 90”, Akio Kawauchi ed., de Gruyter (1992), pp. 447-454.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer Science+Business Media New York

About this chapter

Cite this chapter

Kauffman, L.H. (1993). Combinatorial Recoupling Theory and 3-Manifold Invariants. In: Osborn, H. (eds) Low-Dimensional Topology and Quantum Field Theory. NATO ASI Series, vol 315. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1612-9_1

Download citation

  • DOI: https://doi.org/10.1007/978-1-4899-1612-9_1

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4899-1614-3

  • Online ISBN: 978-1-4899-1612-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics