Abstract
Link polynomial, topological invariant for knots and links, is constructed from an exactly solvable model in statistical mechanics. A general theory consists of two steps. First, representation of the braid group is made from the Boltzmann weights of the exactly solvable model. Second, Markov trace is defined on the braid group representation. Sufficient conditions for the existence of the Markov trace are explicitly given. The knot theory based on exactly solvable models also includes braid-monoid algebra, graphical approach and two-variable extension of link polynomials. In addition, application of the theory to graph-state models is presented.
This is a preview of subscription content, log in via an institution to check access.
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
J.S. Birman: Braids, Links and Mapping Class Groups (Princeton University Press, 1974 ).
L.H. Kauffman: On Knots (Princeton University Press, 1987).
M. Wadati and Y. Akutsu: Prog. Theor. Phys. Suppl. 94 (1988) 1.
Y. Akutsu, T. Deguchi and M. Wadati: in Braid Group, Knot Theory and Statistical Mechanics, ed. C.N. Yang and M.L. Ge (World Scientific Pub., 1989 ).
M. Wadati, T. Deguchi and Y. Akutsu: Phys. Reports (in press).
T. Deguchi, M. Wadati and Y. Akutsu: Adv. Stud. in pure Math. 19 (1989), Kinokuniya-Academic Press.
M. Wadati, T. Deguchi and Y. Akutsu: in Nonlinear Evolution Equations, Integrability and Spectral Methods, ed. A. Fordy (Manchester University Press, 1989 ).
C.N. Yang: Phys. Rev. Lett. 19 (1967) 1312.
R.J. Baxter: Ann. of Phys. 70 (1972) 323.
M. Karowski, H.J. Thun, T.T. Truong and P.H. Weisz: Phys. Lett. 67B (1977) 321.
K. Sogo, M. Uchinami, Y. Akutsu and M. Wadati: Prog. Theor. Phys. 68 (1982) 508.
R.J. Baxter: Exactly Solved Models in Statistical Mechanics (Academic Press,1982)
Y. Akutsu and M. Wadati: J. Phys. Soc.
] Y. Akutsu and M. Wadati: J. Phys. Soc.
Y. Akutsu, T. Deguchi and M. Wadati: J. Phys. Soc. Jpn. 56 (1987) 3464.
Y. Akutsu, T. Deguchi and M. Wadati: J. Phys. Soc. Jpn. 57 (1988) 1173.
T. Deguchi, M. Wadati and Y. Akutsu: J. Phys. Soc. Jpn. 57 (1988) 1905.
T. Deguchi, M. Wadati and Y. Akutsu: J. Phys. Soc. Jpn. 57 (1988) 2921.
V.F.R. Jones: Bull. Amer. Math. Soc. 12 (1985) 103.
P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millett and A. Ocneanu: Bull. Amer. Math. Soc. 12 (1985) 239.
J.H. Przytycki and K.P. Traczyk: Kobe J. Math. 4 (1987) 115.
L. Kauffman: Topology 26 (1987) 395.
V.G. Turaev: Invent. Math. 92 (1988) 527.
J.S. Birman and H. Wenzl: Trans. Amer. Math. Soc. (to appear).
J. Murakami: Osaka J. Math. 24 (1987) 745.
L.H. Kauffman: Statistical Mechanics and the Jones Polynomial, preprint (to appear in Proceedings of 1986 Santa Cruz Conference on the Artin Braid Group )
Y. Akutsu and M. Wadati: Commun. Math. Phys. 117 (1988) 243.
T. Deguchi, Y. Akutsu and M. Wadati: J. Phys. Soc. Jpn. 57 (1988) 757.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer-Verlag Berlin, Heidelberg
About this paper
Cite this paper
Wadati, M., Akutsu, Y., Deguchi, T. (1990). Link Polynomials and Exactly Solvable Models. In: Gu, C., Li, Y., Tu, G., Zeng, Y. (eds) Nonlinear Physics. Research Reports in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-84148-4_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-84148-4_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-52389-5
Online ISBN: 978-3-642-84148-4
eBook Packages: Springer Book Archive