Fibered simple knots

Abstract

We prove that a simple knot in the lens space $L(p,q)$ fibers if and only if its order in homology does not divide any remainder occurring in the Euclidean algorithm applied to the pair $(p,q)$. One corollary is that if $p=m^2$ is a perfect square, then any simple knot of order $m$ fibers, answering a question of Cebanu. More generally, we compute the leading coefficient of the Alexander polynomial of a simple knot, and we describe how to construct a minimum complexity Seifert surface for one. The methods are direct, combinatorial, and geometric.