Knot Theory, Jones Polynomial and Quantum Computing

  • Rūsiņš Freivalds
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3618)


Knot theory emerged in the nineteenth century for needs of physics and chemistry as these needs were understood those days. After that the interest of physicists and chemists was lost for about a century. Nowadays knot theory has made a comeback. Knot theory and other areas of topology are no more considered as abstract areas of classical mathematics remote from anything of practical interest. They have made deep impact on quantum field theory, quantum computation and complexity of computation.


Polynomial Time Quantum Computing Topological Quantum Jones Polynomial Hyperbolic Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Aa 05]
    Aaronson, S.: Guest column: NP-complete problems and physical reality. ACM SIGACT News 36(1), 30–52 (2005)CrossRefGoogle Scholar
  2. [Ad 94]
    Adams, C.C.: The Knot Book. An Elementary Introduction to the Mathematical Theory of Knots. American Mathematical Society, Providence (1994)zbMATHGoogle Scholar
  3. [ADO 92]
    Akutsu, Y., Deguchi, T., Ohtsuki, T.: Invariants of colored links. Journal of Knot Theory Ramifications 1(2), 161–184 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  4. [AKN ]
    Aharonov, D., Kitaev, A., Nisan, N.: Quantum Circuits with Mixed StatesGoogle Scholar
  5. [AJL 05]
    Aharonov, D., Jones, V., Landau, Z.: On the quantum algorithm for approximating the Jones polynomial. Unpublished (2005)Google Scholar
  6. [AL 28]
    Alexander, J.W.: Topological invariants of knots and links. Transactions of American Mathematical Society 30, 275–306 (1928)zbMATHCrossRefGoogle Scholar
  7. [AF 98]
    Ambainis, A., Freivalds, R.: 1-way quantum finite automata: strengths, weaknesses and generalizations. In: Proc. IEEE FOCS 1998, pp. 332–341 (1998) (Also quant-ph/9802062)Google Scholar
  8. [AKV 02]
    Ambainis, A., Kikusts, A., Valdats, M.: On the class of languages recognizable by 1-way quantum finite automataGoogle Scholar
  9. [At 88]
    Atiyah, M.F.: New invariants of three and four dimensional manifolds. In: Proc. Symp. Pure Math.,The mathematical heritage of Herman Weyl, vol. 48, American Mathematical Society, Providence (1988)Google Scholar
  10. [BBBV 97]
    Bennett, C., Bernstein, E., Brassard, G., Vazirani, U.: Strengths and weaknesses of quantum computing. SIAM Journal on Computing 26(5), 1510–1523 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  11. [BV 97]
    Bernstein, E., Vazirani, U.: Quantum complexity theory. SIAM Journal on Computing 26, 1411–1473 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  12. [BP 99]
    Brodsky, A., Pippenger, N.: Characterizations of 1-way quantum finite automata (quant-ph/9903014)Google Scholar
  13. [FKW 02]
    Freedman, M.H., Kitaev, A., Wang, Z.: Simulation of topological field theories by quantum computers. Communications in Mathematical Physics 227(3), 587–603 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  14. [FKLW 01]
    Freedman, M.H., Kitaev, A., Larsen, M.J., Wang, Z.: Topological quantum computation (quant-ph/0101025)Google Scholar
  15. [Jo 89]
    Jones, V.F.R.: On knot invariants related to some statistical mechanical models. Pacific Journal of Mathematics 137(2), 311–334 (1989)zbMATHMathSciNetGoogle Scholar
  16. [Jo 90]
    Jones, V.F.R.: Knot theory and statistical mechanics. Scientific American 263(5), 98–103 (1990)CrossRefMathSciNetGoogle Scholar
  17. [JR 03]
    Jones, V.F.R., Reznikoff, S.A.: Hilbert space representations of the annular Temperley-Lieb algebra,
  18. [Ka 87]
    Kauffman, L.H.: State models and the Jones polynomial. Topology 26(3), 395–407 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  19. [Ka 88]
    Kauffman, L.H.: New invariants in the theory of knots. American Mathematical Monthly 95(3), 195–242 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  20. [Ka 94]
    Kauffman, L.H.: Knot Automata. In: Proc. ISMVL, pp. 328–333 (1994)Google Scholar
  21. [Ka 03]
    Kauffman, L.H.: Review of ”Knots” by Alexei Sossinsky, Harvard University Press (2002), ISBN 0-674-00944-4,
  22. [KL 94]
    Kauffman, L.H., Lins, S.L.: Temperley-Lieb recoupling theory and invariant of 3manifolds. Princeton University Press, Princeton (1994)Google Scholar
  23. [KS 91]
    Kauffman, L.H., Saleur, H.: Free fermions and the Alexander-Conway polynomial. Comm. Math. Phys. 141(2), 293–327 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  24. [Ki 1883]
    Kirkman, T.P.: The enumeration, description and construction of knots with fewer than 10 crossings. Transactions R.Soc. Edinburgh, vol. 32, pp. 281–309 (1883)Google Scholar
  25. [Le 03]
    Levin, L.A.: Polynomial time and extravagant machines, in the tale of one-way machines. Problems of Information Transmission 39(1), 92–103 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  26. [Li 1900]
    Little, C.N.: Non-alternate + - knots. Transactions R.Soc. Edinburgh 39, 771–778 (1900)Google Scholar
  27. [MM 01]
    Murakami, H., Murakami, J.: The colored Jones polynomials and the simplicial volume of a knot. Acta Mathematica 186, 85–104 (2001), Also zbMATHCrossRefMathSciNetGoogle Scholar
  28. [Re 32]
    Werner, K., Reidemeister, F.: Knotentheorie. Eregebnisse der Mathematik und ihrer Grenzgebiete (Alte Folge 0, Band 1, Heft 1). Springer, Berlin (1974) (reprint)Google Scholar
  29. [Si 94]
    Simon, D.: On the power of quantum computation. In: Proc. IEEE FOCS, pp. 116–123 (1994)Google Scholar
  30. [So 02]
    Sossinsky, A.: Knots. Mathematics with a Twist. Harvard University Press, Cambridge (2002)Google Scholar
  31. [Ta 1898]
    Tait, P.G.: On knots I, II, III. In: Scientific papers, vol. 1, pp. 273–347. Cambridge University Press, London (1898)Google Scholar
  32. [Th 1867]
    Thomson, W.: Hydrodynamics. Transactions R.Soc. Edinburgh 6, 94–105 (1867)Google Scholar
  33. [Th 1869]
    Thomson, W.: On vortex motion. Transactions R.Soc. Edinburgh 25, 217–260 (1869)Google Scholar
  34. [Thu 77]
    Thurston, W.P.: The Geometry and Topology of Three-Manifolds. Princeton University Lecture Notes (1977)Google Scholar
  35. [Thu 97]
    Thurston, W.P.: Three-Dimensional Geometry and Topology. Princeton Lecture Notes, vol. 1 (1997)Google Scholar
  36. [Va 90]
    Vassiliev, V.A.: Cohomology of Knot Spaces. In: Arnold, V.I. (ed.) Theory of Singularities and Its Applications, pp. 23–69. Amer. Math. Soc, Providence (1990)Google Scholar
  37. [Wi 89]
    Witten, E.: Quantum field theory and the Jones polynomial. Communications in Mathematical Physics 121(3), 351–399 (1989)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Rūsiņš Freivalds
    • 1
  1. 1.Institute of Mathematics and Computer ScienceUniversity of LatviaRīgaLatvia

Personalised recommendations