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Explicit formulae for two-bridge knot polynomials

Part of: Low-dimensional topology Exponential sums and character sums

Published online by Cambridge University Press:  09 April 2009

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

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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.

Keywords

two-bridge knotAlexander polynomialConvway polynimial

MSC classification

Secondary: 57M25: Knots and links in $S^3$ 11L03: Trigonometric and exponential sums, general
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 78 , Issue 2 , April 2005 , pp. 149 - 166
Copyright
Copyright © Australian Mathematical Society 2005

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