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Alexander polynomials of two-bridge links

Part of: Low-dimensional topology

Published online by Cambridge University Press:  09 April 2009

Taizo Kanenobu
Taizo Kanenobu
Affiliation:
Department of Mathematics kobe UniversityKobe 657, Japan
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Abstract

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We provide an algorithm for calculating the Alexander polynomial of a two-bridge link by putting every two-bridge link in a special type of Conway diagram. Using this algorithm, some necessary conditions for a polynomial to be the Alexander polynomial of a two-bridge link are given, in particular, certain alternating and monotonicity conditions on the coefficients, analogous to corresponding known properties of the reduced Alexander polynomial.

MSC classification

Secondary: 57M25: Knots and links in $S^3$
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 36 , Issue 1 , February 1984 , pp. 59 - 68
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Bailey, J. L., Alexander invariants of links, Ph. D. Thesis (University of British Columbia, 1977).Google Scholar
[2]Conway, J. H., ‘An enumeration of knots and liks, and some their algebraic properties,’ Computational problems in abstract algebra, pp. 329–358 (Pergamon Press, Oxford and New York, 1969).Google Scholar
[3]Fox, R. H., ‘Free differential calculus, II,’ Ann. of Math. 59 (1954), 196–210.CrossRefGoogle Scholar
[4]Hartley, R. I., ‘On two-bridged knot polynomials,’ J. Austral. Math. Soc. 28 (1979), 241–249.CrossRefGoogle Scholar
[5]Kidwell, M. E., ‘Alexander polynomials of links of small order,’ Illinois J. Math. 22 (1978), 459–475.CrossRefGoogle Scholar
[6]Schubert, H., ‘Knoten mit zwei Brücken,’ Math. Z. 65 (1956), 133–170.CrossRefGoogle Scholar
[7]Siebenmann, L., ‘ Exercices sur les noeuds rationnels,’ Orsay, preprint.Google Scholar
[8]Torres, G., ‘On the Alexander polynomials,’ Ann. of Math. 57 (1953), 57–89.CrossRefGoogle Scholar