Abstract
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.
Keywords
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aaronson, S.: Guest column: NP-complete problems and physical reality. ACM SIGACT News 36(1), 30–52 (2005)
Adams, C.C.: The Knot Book. An Elementary Introduction to the Mathematical Theory of Knots. American Mathematical Society, Providence (1994)
Akutsu, Y., Deguchi, T., Ohtsuki, T.: Invariants of colored links. Journal of Knot Theory Ramifications 1(2), 161–184 (1992)
Aharonov, D., Kitaev, A., Nisan, N.: Quantum Circuits with Mixed States
Aharonov, D., Jones, V., Landau, Z.: On the quantum algorithm for approximating the Jones polynomial. Unpublished (2005)
Alexander, J.W.: Topological invariants of knots and links. Transactions of American Mathematical Society 30, 275–306 (1928)
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)
Ambainis, A., Kikusts, A., Valdats, M.: On the class of languages recognizable by 1-way quantum finite automata
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)
Bennett, C., Bernstein, E., Brassard, G., Vazirani, U.: Strengths and weaknesses of quantum computing. SIAM Journal on Computing 26(5), 1510–1523 (1997)
Bernstein, E., Vazirani, U.: Quantum complexity theory. SIAM Journal on Computing 26, 1411–1473 (1997)
Brodsky, A., Pippenger, N.: Characterizations of 1-way quantum finite automata (quant-ph/9903014)
Freedman, M.H., Kitaev, A., Wang, Z.: Simulation of topological field theories by quantum computers. Communications in Mathematical Physics 227(3), 587–603 (2002)
Freedman, M.H., Kitaev, A., Larsen, M.J., Wang, Z.: Topological quantum computation (quant-ph/0101025)
Jones, V.F.R.: On knot invariants related to some statistical mechanical models. Pacific Journal of Mathematics 137(2), 311–334 (1989)
Jones, V.F.R.: Knot theory and statistical mechanics. Scientific American 263(5), 98–103 (1990)
Jones, V.F.R., Reznikoff, S.A.: Hilbert space representations of the annular Temperley-Lieb algebra, http://math.berkeley.edu/vfr/hilbertannular.ps
Kauffman, L.H.: State models and the Jones polynomial. Topology 26(3), 395–407 (1987)
Kauffman, L.H.: New invariants in the theory of knots. American Mathematical Monthly 95(3), 195–242 (1988)
Kauffman, L.H.: Knot Automata. In: Proc. ISMVL, pp. 328–333 (1994)
Kauffman, L.H.: Review of ”Knots” by Alexei Sossinsky, Harvard University Press (2002), ISBN 0-674-00944-4, http://arxiv.org/abs/math.HO/0312168
Kauffman, L.H., Lins, S.L.: Temperley-Lieb recoupling theory and invariant of 3manifolds. Princeton University Press, Princeton (1994)
Kauffman, L.H., Saleur, H.: Free fermions and the Alexander-Conway polynomial. Comm. Math. Phys. 141(2), 293–327 (1991)
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)
Levin, L.A.: Polynomial time and extravagant machines, in the tale of one-way machines. Problems of Information Transmission 39(1), 92–103 (2003)
Little, C.N.: Non-alternate + - knots. Transactions R.Soc. Edinburgh 39, 771–778 (1900)
Murakami, H., Murakami, J.: The colored Jones polynomials and the simplicial volume of a knot. Acta Mathematica 186, 85–104 (2001), Also http://arxiv.org/abs/math/9905075
Werner, K., Reidemeister, F.: Knotentheorie. Eregebnisse der Mathematik und ihrer Grenzgebiete (Alte Folge 0, Band 1, Heft 1). Springer, Berlin (1974) (reprint)
Simon, D.: On the power of quantum computation. In: Proc. IEEE FOCS, pp. 116–123 (1994)
Sossinsky, A.: Knots. Mathematics with a Twist. Harvard University Press, Cambridge (2002)
Tait, P.G.: On knots I, II, III. In: Scientific papers, vol. 1, pp. 273–347. Cambridge University Press, London (1898)
Thomson, W.: Hydrodynamics. Transactions R.Soc. Edinburgh 6, 94–105 (1867)
Thomson, W.: On vortex motion. Transactions R.Soc. Edinburgh 25, 217–260 (1869)
Thurston, W.P.: The Geometry and Topology of Three-Manifolds. Princeton University Lecture Notes (1977)
Thurston, W.P.: Three-Dimensional Geometry and Topology. Princeton Lecture Notes, vol. 1 (1997)
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)
Witten, E.: Quantum field theory and the Jones polynomial. Communications in Mathematical Physics 121(3), 351–399 (1989)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Freivalds, R. (2005). Knot Theory, Jones Polynomial and Quantum Computing. In: Jȩdrzejowicz, J., Szepietowski, A. (eds) Mathematical Foundations of Computer Science 2005. MFCS 2005. Lecture Notes in Computer Science, vol 3618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11549345_2
Download citation
DOI: https://doi.org/10.1007/11549345_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28702-5
Online ISBN: 978-3-540-31867-5
eBook Packages: Computer ScienceComputer Science (R0)