# The Alexander Polynomial of a Rational Link

## TL;DR

This paper explores the Alexander polynomial of rational links, relating boundary terms to diagram features, and provides formulas for its evaluation at specific points, using combinatorial and topological methods.

## Contribution

It introduces a new relationship between the Alexander polynomial's boundary terms and the structure of rational links in standard form, with explicit formulas for special evaluations.

## Key findings

- Boundary terms relate to the number and length of monochromatic twist sites.
- Explicit formula for elta(-1, 0) in terms of twist site crossings.
- Uses Kauffman's clock moves and lattice methods in the proof.

## Abstract

We relate some terms on the boundary of the Newton polygon of the Alexander polynomial $\Delta(x,y)$ of a rational link to the number and length of monochromatic twist sites in a particular diagram that we call the standard form. Normalize $\Delta(x,y)$ so that no $x^{-1}$ or $y^{-1}$ terms appear, but $x^{-1}\Delta(x,y)$ and $y^{-1}\Delta(x,y)$ have negative exponents, and so that terms of even total degree are positive and terms with odd total degree are negative. If the rational link has a reduced alternating diagram with no self crossings, then $\Delta(-1, 0) = 1$. If the standard form of the rational link has $m$ monochromatic twist sites, and the $j^{\textrm{th}}$ monochromatic twist site has $\hat{q}_j$ crossings, then $\Delta(-1, 0) = \prod_{j=1}^{m}(\hat{q}_j+1)$. Our proof employs Kauffman's clock moves and a lattice for the terms of $\Delta(x,y)$ in which the $y$-power cannot decrease.