Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T01:44:35.587Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantum SU(2)-invariants dominate Casson's SU(2)-invariant

Published online by Cambridge University Press:  24 October 2008

Hitoshi Murakami
Hitoshi Murakami
Affiliation:
Department of Mathematics, Osaka City University, Sugimoto, Sumiyoshi-ku, Osaka, 558, Japan
Get access Rights & Permissions [Opens in a new window]

Extract

In 1988, E. Witten proposed an array of invariants (quantum G-invariants) for a 3-manifold associated with a compact simple Lie group G based on the quantum field theory [22]. N. Reshetikhin and V. Turaev [20] gave a mathematical approach to these invariants for G = SU(2) by using representations of the quantum group Uq(sl(2, ℂ)). See also [2, 11, 12, 14, 15, 16] for other approaches to these invariants: [16] may be the simplest way (for knot theorists) to understand the existence of these invariants.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 115 , Issue 2 , March 1994 , pp. 253 - 281
Copyright
Copyright © Cambridge Philosophical Society 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Akbulut, S. and McCarthy, J. D.. Casson's invariant for oriented homology 3-spheres. An exposition. Mathematical Notes 36. (Princeton University Press, 1990).CrossRefGoogle Scholar
[2]Blanchet, C., Habegger, N., Masbaum, G., and Vogel, P.. Three-manifold invariants derived from the Kauffman bracket. Topology 31 (1992), 685–699.CrossRefGoogle Scholar
[3]Conway, J. H.. An enumeration of knots and links, and some of their algebraic properties. In Computational Problems in Abstract Algebra (Pergamon Press, 1969).Google Scholar
[4]Hoste, J., A formula for Casson's invariant. Trans. Amer. Math. Soc. 297 (1986), 547–562.Google Scholar
[5]Ireland, K. and Rosen, M.. A Classical Introduction to Modern Number Theory, Second Edition (Springer-Verlag, 1990).CrossRefGoogle Scholar
[6]Jones, V. F. R.. A polynomial invariant for knots via von Neumann algebras. Bull. Amer. Math. Soc. 12 (1985), 103–111.CrossRefGoogle Scholar
[7]Jones, V. F. R.. Hecke algebra representations of braid groups and link polynomials. Ann. Math. 126 (1987), 335–388.CrossRefGoogle Scholar
[8]Kauffman, L. H.. The Conway polynomial. Topology 20 (1981), 101–108.CrossRefGoogle Scholar
[9]Kauffman, L. H.. State models and the Jones polynomial. Topology 26 (1987), 395–407.CrossRefGoogle Scholar
[10]Kirby, R. C.. A calculus for framed links in S 3. Invent. math. 45 (1978), 35–56.CrossRefGoogle Scholar
[11]Kirby, R. and Melvin, P.. The 3-manifold invariants of Witten and Reshetikhin–Turaev for sl(2, ℂ). Invent. math. 105 (1991), 473–545.CrossRefGoogle Scholar
[12]Kohno, T.. Topological invariants for 3-manifolds using representations of mapping class groups. Topology 31 (1992), 203–230.CrossRefGoogle Scholar
[13]Lickorish, W. B. R.. A representation of orientable combinatorial 3-manifolds. Ann. Math. 76 (1962), 531–540.CrossRefGoogle Scholar
[14]Lickorish, W. B. R.. Three-manifolds and the Temperley–Lieb algebra. Math. Ann. 290 (1991), 657–670.CrossRefGoogle Scholar
[15]Lickorish, W. B. R.. Calculations with the Temperley–Lieb algebra. Comm. Math. Helv., 67, (1992), 571–591.CrossRefGoogle Scholar
[16]Lickorish, W. B. R.. The skein method for three-manifold invariants. J. Knot Theory and Its Ramification, 2, (1993), 171–194.CrossRefGoogle Scholar
[17]Lickorish, W. B. R. and Millett, K. C.. The reversing result for the Jones polynomial. Pacific J. Math. 124 (1986), 173–176.CrossRefGoogle Scholar
[18]Morton, H. R.. The Jones polynomial for unoriented links. Quart. J. Math. 37 (1986), 55–60.CrossRefGoogle Scholar
[19]Murakami, H.. On derivatives of the Jones polynomial. Kobe J. Math. 3 (1986), 61–64.Google Scholar
[20]Reshetikhin, N. Yu. and Turaev, V. G.. Invariants of 3-manifolds via link polynomials and quantum groups. Invent. math. 103 (1991), 547–597.CrossRefGoogle Scholar
[21]Walker, K.. An extension of Casson's invariant, Annals of Mathematics Studies. 126 (Princeton University Press, 1992).CrossRefGoogle Scholar
[22]Witten, E.. Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121 (1989), 351–399.CrossRefGoogle Scholar