Embedding infinite cyclic covers of knot spaces into 3-space

2005/05/11 by Boju Jiang, Jiang, Boju, Yi Ni +5
Mathematics · #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.DS #math.GT #msc:37E99 #msc:57M10 #msc:57M25 #msc:57N30

paper · pdf · doi:10.48550/arxiv.math/0505206

24 pages, 9 figures

arxiv created 2005/05/11 · arxiv updated 2009/12/01

Abstract

We say a knot k in the 3-sphere \mathbb S³ has \it Property IE if the infinite cyclic cover of the knot exterior embeds into \mathbb S³. Clearly all fibred knots have Property IE. There are infinitely many non-fibred knots with Property IE and infinitely many non-fibred knots without property IE. Both kinds of examples are established here for the first time. Indeed we show that if a genus 1 non-fibred knot has Property IE, then its Alexander polynomial Δ_k(t) must be either 1 or 2t²-5t+2, and we give two infinite families of non-fibred genus 1 knots with Property IE and having Δ_k(t)=1 and 2t²-5t+2 respectively. Hence among genus one non-fibred knots, no alternating knot has Property IE, and there is only one knot with Property IE up to ten crossings. We also give an obstruction to embedding infinite cyclic covers of a compact 3-manifold into any compact 3-manifold.

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