Reducible handle additions to weakly reducible Heegaard splittings

We show that there are infinitely many two component links in S3 whose complements have weakly reducible and irreducible non-minimal genus Heegaard splittings, yet the construction given in the theorem of Casson and Gordon does not produce an essential closed surface. The situation for manifolds with a single boundary component is still unresolved though we obtain partial results regarding manifolds with a non-minimal genus weakly reducible and irreducible Heegaard splitting.

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Weakly reducible Heegaard splittings of Seifert fibered spaces

Topology and its Applications ◽

10.1016/s0166-8641(98)00091-1 ◽

2000 ◽

Vol 100 (2-3) ◽

pp. 219-222 ◽

Cited By ~ 2

Author(s):

Jennifer Schultens

Keyword(s):

Heegaard Splittings ◽

Seifert Fibered Spaces ◽

Weakly Reducible

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Local detection of strongly irreducible Heegaard splittings

Topology and its Applications ◽

10.1016/s0166-8641(97)00184-3 ◽

1998 ◽

Vol 90 (1-3) ◽

pp. 135-147 ◽

Cited By ~ 28

Author(s):

Martin Scharlemann

Keyword(s):

Heegaard Splittings ◽

Strongly Irreducible ◽

Local Detection

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Local uniqueness of certain geodesics related to Heegaard splittings

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216518420038 ◽

2018 ◽

Vol 27 (09) ◽

pp. 1842003

Author(s):

Liang Liang ◽

Fengling Li ◽

Fengchun Lei ◽

Jie Wu

Keyword(s):

Heegaard Splitting ◽

Sufficient Condition ◽

Heegaard Splittings ◽

Local Uniqueness ◽

Uniqueness Property

Suppose [Formula: see text] is a Heegaard splitting and [Formula: see text] is an essential separating disk in [Formula: see text] such that a component of [Formula: see text] is homeomorphic to [Formula: see text], [Formula: see text]. In this paper, we prove that if there is a locally complicated simplicial path in [Formula: see text] connecting [Formula: see text] to [Formula: see text], then the geodesic connecting [Formula: see text] to [Formula: see text] is unique. Moreover, we give a sufficient condition such that [Formula: see text] is keen and the geodesic between any pair of essential disks on the opposite sides has local uniqueness property.

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