Fibered simple knots

TL;DR

This paper characterizes when simple knots in lens spaces fiber, linking it to the Euclidean algorithm, and provides methods to compute Alexander polynomials and construct minimal Seifert surfaces.

Contribution

It establishes a precise criterion for fibering of simple knots in lens spaces based on homology and Euclidean algorithm remainders, and offers new computational tools.

Findings

Simple knots in lens spaces fiber iff their homology order does not divide any Euclidean algorithm remainder.

For perfect square p=m^2, all simple knots of order m fiber.

Provides formulas for Alexander polynomial coefficients and methods for minimal Seifert surface construction.

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.