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
  Tokyo 187-8577

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