Refilling meridians in a genus 2 handlebody complement. (English) Zbl 1177.57019

Boileau, Michel (ed.) et al., The Zieschang Gedenkschrift. Coventry: Geometry & Topology Publications. Geometry and Topology Monographs 14, 451-475 (2008).
Let \(W\) be a genus two handlebody embedded in a compact orientable 3-manifold \(M\) and let \(\alpha\) and \(\beta\) be two not necessarily disjoint essential properly embedded disks in \(W\), called meridian disks. Then \(W\setminus \eta(\alpha)\) (resp. \(W\setminus \eta(\beta)\)) is a regular neighborhood of a knot or link \(L[\alpha]\) (resp. \(L[\beta]\)) in \(M\). Let \(M[\alpha]\) and \(M[\beta]\) be the manifolds obtained from \(M\setminus W\) by restoring neighborhoods of the meridians \(\alpha\) and \(\beta\). The main question of the paper is to establish under what hypothesis \(M[\alpha]\) and \(M[\beta]\) are both reducible and/or \(\delta\)-reducible. The author states the following conjecture: if \((M,W)\) is an admissible pair then \(W\) is unknotted and \(M=S^3\), or at least one of \(M[\alpha]\) and \(M[\beta]\) is both irreducible and \(\delta\)-irreducible, or \(\alpha\) and \(\beta\) are aligned in \(M\). This conjecture is proved to be true in three important special cases: when \(M\setminus W\) is \(\delta\)-reducible, when \(|\partial \alpha \cap \partial \beta|\leq 4\), and when both \(\alpha\) and \(\beta\) are separating meridians. Conversely, it is also shown that if \(\alpha\) and \(\beta\) are aligned meridians in \(W\), then there is an unknotted embedding of \(W\) in \(\mathbb S^3\) (hence in any 3-manifold) so that each of \(L[\alpha]\) and \(L[\beta]\) is either the unknot or a split link.
For the entire collection see [Zbl 1135.00012].


57N10 Topology of general \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
Full Text: arXiv