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The skein polynomial of a planar star product of two links

Published online by Cambridge University Press:  28 June 2011

Kunio Murasugi  and
Jozef H. Przytycki
Kunio Murasugi
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Canada M5S 1A1
Jozef H. Przytycki
Affiliation:
Department of Mathematics, Warsaw University, Warsaw, Poland 00901
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Abstract

If PL(v,z) = Σbi(v)zi is the skein polynomial of a link L, and D = D1 * D2 is the diagram which is a planar star (Murasugi) product of D1 and D2 then bϕ(D)(v) = bϕ(D1)·bϕ(D2)(v) where ϕ(D) = n(D)– (s(D) – 1) and n(D) denotes the number of crossings of D, and s(D) is the number of Seifert circles of D.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 106 , Issue 2 , September 1989 , pp. 273 - 276
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

[1] Boileau, M. and Weber, C.. Le problème de J. Milnor sur le nombre gordien des noeuds algébraiques. Enseign. Math. (2) 31 (1983), 49–98.Google Scholar
[2] Cromwell, P. R.. Homogeneous links. (Preprint, 1987.)Google Scholar
[3] Freyd, P., Yetter, D., Hoste, J., Lickorish, W. B. R., Millett, K. and Ocneanu, A.. A new polynomial invariant of knots and links. Bull. Amer. Math. Soc. 12 (1985), 239–249.CrossRefGoogle Scholar
[4] Kobayashi, K. and Kodama, K.. On the degzPL(v, z) for plumbing diagrams and oriented arborescent links. Kobe J. Math. 5 (1988), 221–232.Google Scholar
[5] Milnor, J.. Singular Points of Complex Hypersurfaces. Ann. of Math. Studies no. 61 (Princeton University Press, 1968).Google Scholar
[6] Morton, H. R.. Seifert circles and knot polynomials. Math. Proc. Cambridge Philos. Soc. 99 (1986), 107–109.CrossRefGoogle Scholar
[7] Murasugi, K.. On alternating knots. Osaka Math. J. 12 (1960), 277–303.Google Scholar
[8] Murasugi, K.. On a certain numerical invariant of link types. Trans. Amer. Math. Soc. 117 (1965), 387–422.CrossRefGoogle Scholar
[9] Przytycki, J. H.. Survey on recent invariants in classical knot theory. (Preprint, 1986.)Google Scholar
[10] Przytycki, J. H. and Traczyk, P.. Invariants of links of Conway type. Kobe J. Math. 4 (1987), 115–139.Google Scholar
[11] Przytycki, J. H. and Traczyk, P.. Conway algebras and skein equivalence of links. (Preprint, 1985.)Google Scholar