scholarly journals Manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings

2009 ◽  
Vol 17 (4) ◽  
pp. 637-649 ◽  
Author(s):  
Tsuyoshi Kobayashi ◽  
Yo’av Rieck
Keyword(s):  
Heegaard Splittings ◽  
Minimal Genus ◽  
Strongly Irreducible ◽  
Weakly Reducible

CLOSED ESSENTIAL SURFACES AND WEAKLY REDUCIBLE HEEGAARD SPLITTINGS IN MANIFOLDS WITH BOUNDARY

2004 ◽  
Vol 13 (06) ◽  
pp. 829-843 ◽  
Author(s):  
YOAV MORIAH ◽  
ERIC SEDGWICK
Keyword(s):  
Boundary Component ◽  
Closed Surface ◽  
Heegaard Splitting ◽  
Heegaard Splittings ◽  
Manifolds With Boundary ◽  
Minimal Genus ◽  
Essential Surfaces ◽  
Two Component ◽  
Single Boundary ◽  
Weakly Reducible

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.

scholarly journals A distance for circular Heegaard splittings

2019 ◽  
Vol 28 (09) ◽  
pp. 1950059
Author(s):  
Kevin Lamb ◽  
Patrick Weed
Keyword(s):  
Critical Points ◽  
Morse Function ◽  
Heegaard Splitting ◽  
Heegaard Splittings ◽  
Seifert Surface ◽  
Singular Foliation ◽  
Incompressible Surfaces ◽  
Minimal Genus ◽  
Seifert Surfaces ◽  
Index 2

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.

Minimal partitions for critical Heegaard splittings

2020 ◽  
Vol 29 (04) ◽  
pp. 2050023
Author(s):  
J. H. Lee ◽  
T. Saito
Keyword(s):  
Critical Surface ◽  
Heegaard Splittings ◽  
Minimal Genus ◽  
Heegaard Surface

In this paper, we define the minimality of a partition for a critical Heegaard surface. The standard minimal genus Heegaard surface of [Formula: see text], which is known to be critical, admits a minimal partition. Moreover, we give an example of a critical surface that admits both a minimal partition and a non-minimal partition.

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