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Jones polynomial invariants for knots and satellites

Published online by Cambridge University Press:  24 October 2008

H. R. Morton  and
P. Strickland
H. R. Morton
Affiliation:
Department of Pure Mathematics, University of Liverpool, PO Box 147, Liverpool L69 3BX
P. Strickland
Affiliation:
Department of Pure Mathematics, University of Liverpool, PO Box 147, Liverpool L69 3BX
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Abstract

Results of Kirillov and Reshetikhin on constructing invariants of framed links from the quantum group SU(2)q are adapted to give a simple formula relating the invariants for a satellite link to those of the companion and pattern links used in its construction. The special case of parallel links is treated first. It is shown as a consequence that any SU(2)q-invariant of a link L is a linear combination of Jones polynomials of parallels of L, where the combination is determined explicitly from the representation ring of SU(2). As a simple illustration Yamada's relation between the Jones polynomial of the 2-parallel of L and an evaluation of Kauffman's polynomial for sublinks of L is deduced.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 109 , Issue 1 , January 1991 , pp. 83 - 103
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Drinfeld, V. G.. Quantum groups. In Proceedings International Congress of Mathematicians, Berkeley, 1986 (American Mathematical Society, 1987), pp. 798–820.Google Scholar
[2]Fox, R. H.. Free differential calculus V: the Alexander matrices re-examined. Ann. of Math. 71 (1960), 408–422.CrossRefGoogle Scholar
[3]Jimbo, M.. A q-difference analogue of U(g) and the Yang–Baxter equations. Lett. Math. Phys. 10 (1985), 63–69.CrossRefGoogle Scholar
[4]Jones, V. F. R.. A polynomial invariant for knots via von Neumann algebras. Bull. Amer. Math. Soc. 12 (1985), 103–111.CrossRefGoogle Scholar
[5]Joyal, A. and Street, R.. Braided monoidal categories. (Macquarie Math. Reports no. 860081, 1986.)Google Scholar
[6]Kluitmann, P.. Notes on knot invariants. (Liverpool, 03 1986.)Google Scholar
[7]Kirillov, A. N. and Reshetikhin, N. Yu.. Representations of the algebra Uq(sl 2), q-orthogonal polynomials and invariants of links. In Infinite-Dimensional Lie Algebras and Groups, Adv. Ser. in Math. Phys. no. 7 (Publisher, 1988) (editor Kac, V. G.), pp. 285–339.Google Scholar
[8]Murakami, J.. The parallel version of link invariants. Osaka J. Math. 26 (1989), 1–55.Google Scholar
[9]Morton, H. R.. Satellites and knot invariants. (Notes of a talk given to the London Mathematical Society, 1989.)Google Scholar
[10]Morton, H. R. and Short, H. B.. The 2-variable polynomial of cable knots. Math. Proc. Cambridge Philos. Soc. 101 (1987), 267–278.CrossRefGoogle Scholar
[11]Morton, H. R. and Traczyk, P.. The Jones polynomials of satellite links and mutants. In Braids, Contemp. Math. no. 78 (American Mathematical Society, 1988), pp. 575–585.CrossRefGoogle Scholar
[12]Reshetikhin, N. Yu.. Quantum universal enveloping algebras, the Yang–Baxter equation and invariants of links I. (LOMI preprint E-4–87, Leningrad 1988.)Google Scholar
[13]Reshetikhin, N. Yu.. Quantum universal enveloping algebras, the Yang–Baxter equation and invariants of links II. (LOMI preprint E-17–87, Leningrad 1988.)Google Scholar
[14]Reshetikhin, N. Yu. and Turaev, V. G.. Ribbon graphs and their invariants derived from quantum groups. (Preprint, MSRI 1989.)Google Scholar
[15]Rosso, M.. Représentations irréductibles de dimension finie du q-analogue de l'algèbre enveloppante d'une algèbre de Lie simple. C. R. Acad. Sci. Paris Ser. I Math. 305 (1987), 587–590.Google Scholar
[16]Turaev, V. G.. The Yang–Baxter equation and invariants of links. Invent. Math. 92 (1988), 527–553.CrossRefGoogle Scholar
[17]Turaev, V. G.. Operator invariants of tangles and R-matrices. Izv. Akad. Nauk SSSR 53 (1989), 1073–1107.Google Scholar
[18]Turaev, V. G.. The category of oriented tangles and its representations. Funktsional. Anal. i Prilozhen. 23 (1989), 93–94.Google Scholar
[19]Yamada, S.. An operator on regular isotopy invariants of links. Topology 28 (1989), 369–377.CrossRefGoogle Scholar