On the topological invariance of Murasugi special components of an alternating link

Abstract

Let $L$ be an unsplittable, prime, oriented, alternating link type in $S^3$. Let $D$ be a reduced alternating diagram representing $L$. We define the Murasugi atoms of $D$ as the oriented link types represented by the prime factors of the Murasugi special components of $D$. We prove (an invariance theorem) that the collection of Murasugi atoms depends only on $L$ and not on $D$. This has the following corollary. Let $L$ be as above and assume that $L$ is achiral. Write its HOMFLY polynomial as $P_{L}(v,z)\,{=}\,\sum_{m}^{M} b_{j}(v) z^j$. Then $b_{M}(v)\,{=}{\pm}\, \beta(v) \beta(v^{-1})$ for some polynomial $\beta(v) \in\mathbb{Z}[v, v^{-1}]$. As a consequence, the leading coefficient of the Conway polynomial of $L$ is a square (up to sign).