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Explicit formulae for two-bridge knot polynomials

Published online by Cambridge University Press:  09 April 2009

Shinji Fukuhara
Affiliation:
Department of MathematicsTsuda CollegeTsuda-machi 2-1-1 Kodaira-shi Tokyo 187-8577Japan e-mail: fukuhara@tsuda.ac.jp
Corresponding
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Abstract

A two-bridge knot (or link) can be characterized by the so-called Schubert normal form Kp, q where p and q are positive coprime integers. Associated to Kp, q there are the Conway polynomial ▽kp, q(z) and the normalized Alexander polynomial Δkp, q(t). However, it has been open problem how ▽kp, q(z) and Δkp, q(t) are expressed in terms of p and q. In this note, we will give explicit formulae for the Conway polynomials and the normalized Alexander polynomials in the case of two-bridge knots and links. This is done using elementary number theoretical functions in p and q.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2005

References

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