Explicit formulae for two-bridge knot polynomials

J. Aust. Math. Soc. 78 (2005), 149-166

Explicit formulae for two-bridge knot polynomials

Shinji Fukuhara
Department of Mathematics
Tsuda College
Tsuda-machi 2-1-1
Kodaira-shi
Tokyo 187-8577
Japan
fukuhara@tsuda.ac.jp

Abstract

A two-bridge knot (or link) can be characterized by the so-called Schubert normal form $K_{p,q}$ where

and

are positive coprime integers. Associated to $K_{p,q}$ there are the Conway polynomial $\nabla_{K_{p,q}}(z)$ and the normalized Alexander polynomial $\Delta_{K_{p,q}}(t)$ . However, it has been open problem how $\nabla_{K_{p,q}}(z)$ and $\Delta_{K_{p,q}}(t)$ are expressed in terms of

and

. 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

and

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