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Twisting formulae of the Jones polynomial

Published online by Cambridge University Press:  24 October 2008

Yoshiyuki Yokota
Yoshiyuki Yokota
Affiliation:
Department of Mathematics, School of Science and Engineering, Waseda University, Ohkubo, Shinjuku, Tokyo, 169, Japan
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Extract

Let be an oriented knot in S3 and a solid torus endowed with a preferred framing which contains in its interior. By ρ and λ we denote the wrapping and winding numbers of in respectively. That is, they are the geometric and algebraic intersection numbers of and a meridian disk of . For an integer μ, let τμ be an orientation-preserving homeomorphism of satisfying τμ(m) = m and τμ(l) = l + μm in H1(∂), where (m, l) is a meridian-longitude pair of . We call τμ(), denoted by μ, the knot obtained from by μ-twisting along .

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 3 , November 1991 , pp. 473 - 482
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Hitt, L. R. and Silver, D. R.. Stallings twists and the Jones polynomial. Preprint (1988).Google Scholar
[2]Jones, V. F. R.. Hecke algebra representations of braid groups and link polynomials. Ann. of Math. (2) 126 (1987), 335–388.CrossRefGoogle Scholar
[3]Kauffman, L. H.. State models and the Jones polynomial. Topology 26 (1987), 395–407.CrossRefGoogle Scholar
[4]Kouno, M., Motegi, K. and Shibuya, T.. Twisting and knot types. Preprint.Google Scholar
[5]Kouno, M., Motegi, K. and Shibuya, T.. Twisting of knots. Preprint.Google Scholar
[6]Morton, H. R.. Fibred knots with a given Alexander polynomial. Enseign. Math. 31 (1983), 206–222.Google Scholar
[7]Morton, H. R. and Strickland, P. M.. Jones polynomial invariants for knots and satellites. Math. Proc. Cambridge Philos. Soc. 109 (1991), 83–103.CrossRefGoogle Scholar
[8]Morton, H. R. and Traczyk, P.. Knots and algebras. Contribuciones Matematicas (1990).Google Scholar
[9]Murakami, J.. The Kauffman polynomial of links and representation theory. Osaka J. Math. 24 (1987), 745–758.Google Scholar
[10]Murakami, J.. The parallel version of polynomial invariants of links. Osaka J. Math. 26 (1989), 1–55.Google Scholar
[11]Whitehead, J. H. C.. On doubled knots. J. London Math. Soc. 12 (1937), 63–71.CrossRefGoogle Scholar