Decomposing Heegaard splittings along separating incompressible surfaces in 3-manifolds

For a knot [Formula: see text], its exterior [Formula: see text] has a singular foliation by Seifert surfaces of [Formula: see text] derived from a circle-valued Morse function [Formula: see text]. When [Formula: see text] is self-indexing and has no critical points of index 0 or 3, the regular levels that separate the index-1 and index-2 critical points decompose [Formula: see text] into a pair of compression bodies. We call such a decomposition a circular Heegaard splitting of [Formula: see text]. We define the notion of circular distance (similar to Hempel distance) for this class of Heegaard splitting and show that it can be bounded under certain circumstances. Specifically, if the circular distance of a circular Heegaard splitting is too large: (1) [Formula: see text] cannot contain low-genus incompressible surfaces, and (2) a minimal-genus Seifert surface for [Formula: see text] is unique up to isotopy.

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Separating incompressible surfaces and stabilizations of Heegaard splittings

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004104007790 ◽

2004 ◽

Vol 137 (3) ◽

pp. 633-643 ◽

Cited By ~ 24

Author(s):

TSUYOSHI KOBAYASHI ◽

RUIFENG QIU ◽

YO'AV RIECK ◽

SHICHENG WANG

Keyword(s):

Heegaard Splittings ◽

Incompressible Surfaces

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Incompressible surfaces in 2-bridge knot complements

Inventiones mathematicae ◽

10.1007/bf01388971 ◽

1985 ◽

Vol 79 (2) ◽

pp. 225-246 ◽

Cited By ~ 113

Author(s):

A. Hatcher ◽

W. Thurston

Keyword(s):

Incompressible Surfaces ◽

Knot Complements

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One-sided incompressible surfaces in Seifert fibered spaces

Topology and its Applications ◽

10.1016/0166-8641(86)90032-5 ◽

1986 ◽

Vol 23 (2) ◽

pp. 103-116 ◽

Cited By ~ 7

Author(s):

Charles Frohman

Keyword(s):

Incompressible Surfaces ◽

Seifert Fibered Spaces

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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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