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Jones polynomial invariants for knots and satellites

Published online by Cambridge University Press:  24 October 2008

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

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
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. 798820.Google Scholar
[2]Fox, R. H.. Free differential calculus V: the Alexander matrices re-examined. Ann. of Math. 71 (1960), 408422.CrossRefGoogle Scholar
[3]Jimbo, M.. A q-difference analogue of U(g) and the Yang–Baxter equations. Lett. Math. Phys. 10 (1985), 6369.CrossRefGoogle Scholar
[4]Jones, V. F. R.. A polynomial invariant for knots via von Neumann algebras. Bull. Amer. Math. Soc. 12 (1985), 103111.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. 285339.Google Scholar
[8]Murakami, J.. The parallel version of link invariants. Osaka J. Math. 26 (1989), 155.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), 267278.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. 575585.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), 587590.Google Scholar
[16]Turaev, V. G.. The Yang–Baxter equation and invariants of links. Invent. Math. 92 (1988), 527553.CrossRefGoogle Scholar
[17]Turaev, V. G.. Operator invariants of tangles and R-matrices. Izv. Akad. Nauk SSSR 53 (1989), 10731107.Google Scholar
[18]Turaev, V. G.. The category of oriented tangles and its representations. Funktsional. Anal. i Prilozhen. 23 (1989), 9394.Google Scholar
[19]Yamada, S.. An operator on regular isotopy invariants of links. Topology 28 (1989), 369377.CrossRefGoogle Scholar