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=m2 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.
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