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.

Identifying non-pseudo-alternating knots by using the free factor property of minimal genus Seifert surfaces

2021 ◽  
Author(s):  
Keisuke Himeno ◽  
Masakazu Teragaito
Keyword(s):  
Seifert Surface ◽  
Major Difficulty ◽  
Flat Surfaces ◽  
Minimal Genus ◽  
Pretzel Knots ◽  
Seifert Surfaces ◽  
Manifold Theory ◽  
Knots And Links

Pseudo-alternating knots and links are defined constructively via their Seifert surfaces. By performing Murasugi sums of primitive flat surfaces, such a knot or link is obtained as the boundary of the resulting surface. Conversely, it is hard to determine whether a given knot or link is pseudo-alternating or not. A major difficulty is the lack of criteria to recognize whether a given Seifert surface is decomposable as a Murasugi sum. In this paper, we propose a new idea to identify non-pseudo-alternating knots. Combining with the uniqueness of minimal genus Seifert surface obtained through sutured manifold theory, we demonstrate that two infinite classes of pretzel knots are not pseudo-alternating.

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.

KNOTTED SEIFERT SURFACES AND UNKNOTTED SEIFERT SURFACES

2008 ◽  
Vol 17 (02) ◽  
pp. 141-155
Author(s):  
YUKIHIRO TSUTSUMI
Keyword(s):  
Seifert Surface ◽  
Minimal Genus ◽  
Seifert Surfaces ◽  
Genus One

It is known that free genus one knots do not admit Seifert surfaces with hyperbolic exteriors. In this paper, for any integer g ≥ 2, we exhibit a knot of genus g which bounds a minimal genus Seifert surface with hyperbolic exterior and a minimal genus free Seifert surface.

Local uniqueness of certain geodesics related to Heegaard splittings

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.

scholarly journals SEIFERT SURFACES, COMMUTATORS AND VASSILIEV INVARIANTS

2007 ◽  
Vol 16 (10) ◽  
pp. 1295-1329
Author(s):  
E. KALFAGIANNI ◽  
XIAO-SONG LIN
Keyword(s):  
Fundamental Group ◽  
Lower Central Series ◽  
Central Series ◽  
Seifert Surface ◽  
Vassiliev Invariants ◽  
Seifert Surfaces

We show that the Vassiliev invariants of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its fundamental group. We also conjecture a characterization of knots whose invariants of all orders vanish in terms of their Seifert surfaces.

scholarly journals Decomposing Heegaard splittings along separating incompressible surfaces in 3-manifolds

2019 ◽  
Vol 264 ◽  
pp. 21-26
Author(s):  
Kazuhiro Ichihara ◽  
Makoto Ozawa ◽  
J. Hyam Rubinstein
Keyword(s):  
Heegaard Splittings ◽  
Incompressible Surfaces

scholarly journals Some examples of minimally degenerate Morse functions

1987 ◽  
Vol 30 (2) ◽  
pp. 289-293 ◽  
Author(s):  
Frances Kirwan
Keyword(s):  
Riemannian Manifold ◽  
Critical Points ◽  
Morse Function ◽  
Compact Riemannian Manifold ◽  
Connected Components ◽  
Morse Inequalities ◽  
Poincaré Polynomial ◽  
Morse Functions ◽  
Image Position

Let X be a compact Riemannian manifold. If f:X→ℝ is a nondegenerate Morse function in the sense of Bott [2] then one has Morse inequalities which can be expressed in the formwhere Pt(X) is the Poincaré polynomial Σtidim Hi(X;ℚ of X ann {Cβ|β ∈B} are the connected components of the set of critical points for f For any polynomial Q(t)∈ℤ[t] we write Q(t)≧0 if all the coefficients of Q are nonnegative.

scholarly journals Umbilical Submanifolds and Morse Functions

1972 ◽  
Vol 48 ◽  
pp. 197-201 ◽  
Author(s):  
Katsumi Nomizu ◽  
Lucio Rodríguez
Keyword(s):  
Euclidean Space ◽  
Differentiable Function ◽  
Critical Points ◽  
Morse Function ◽  
Differentiable Manifold ◽  
Morse Functions

Let Mn be a differentiable manifold (of class C∞). By a Morse function on Mn we mean a differentiable function whose critical points are all non-degenerate. If f is an immersion of Mn into a Euclidean space Rm, we may obtain Morse functions on Mn in the following way.

SEIFERT SURFACES IN KNOT COMPLEMENTS

2007 ◽  
Vol 16 (08) ◽  
pp. 1053-1066 ◽  
Author(s):  
ENSIL KANG
Keyword(s):  
Compact Surface ◽  
Seifert Surface ◽  
Normal Surface ◽  
Incompressible Surface ◽  
Ideal Triangulation ◽  
Seifert Surfaces ◽  
Knot Complements

In the ordinary normal surface for a compact 3-manifold, any incompressible, ∂-incompressible, compact surface can be moved by an isotopy to a normal surface [9]. But in a non-compact 3-manifold with an ideal triangulation, the existence of a normal surface representing an incompressible surface cannot be guaranteed. The figure-8 knot complement is presented in a counterexample in [12]. In this paper, we show the existence of normal Seifert surface under some restriction for a given ideal triangulation of the knot complement.

RECTANGLE CONDITION FOR COMPRESSION BODIES AND 2-FOLD BRANCHED COVERINGS

2012 ◽  
Vol 21 (08) ◽  
pp. 1250078
Author(s):  
JUNGSOO KIM ◽  
JUNG HOON LEE
Keyword(s):  
Heegaard Splitting ◽  
Level Surface ◽  
Heegaard Splittings ◽  
Branched Coverings ◽  
Link Complement ◽  
Strong Irreducibility

We give the rectangle condition for strong irreducibility of Heegaard splittings of 3-manifolds with non-empty boundary. We apply this to a generalized Heegaard splitting of 2-fold covering of S3 branched along a link. The condition implies that any thin meridional level surface in the link complement is incompressible. We also show that the additivity of width holds for a composite knot satisfying the condition.

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