Small knots and large handle additions

2004/02/08 by Ruifeng Qiu, Qiu, Ruifeng, Shicheng Wang +1
Mathematics · #57M10 #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:57M10

paper · pdf · doi:10.48550/arxiv.math/0402120

25 pages, 14 figures

arxiv created 2004/02/08 · arxiv updated 2009/12/01

Abstract

We construct a hyperbolic 3-manifold M (with ∂ M totally geodesic) which contains no essential closed surfaces, but for any even integer g> 0 there are infinitely many separating slopes r on ∂ M so that M[r], the 3-manifold obtained by attaching 2-handle to M along r, contains an essential separating closed surface of genus g and is still hyperbolic. The result contrasts sharply with those known finiteness results for the cases g=0,1. Our 3-manifold M is the complement of a simple small knot in a handlebody.

Related

• Handle additions producing essential surfaces
• Small curvature surfaces in hyperbolic 3-manifolds
• Totally geodesic boundaries of knot complements
• Knots with only two strict essential surfaces
• On finiteness of the number of boundary slopes of immersed surfaces in 3-manifolds