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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].

MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M50 General geometric structures on low-dimensional manifolds
Keywords:
3-manifold; genus two handlebody; knot; link.
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