scholarly journals Uniqueness of minimal genus Seifert surfaces for links

1989 ◽  
Vol 33 (3) ◽  
pp. 265-279 ◽  
Author(s):  
Tsuyoshi Kobayashi
Keyword(s):  
Minimal Genus ◽  
Seifert Surfaces

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.

SURGERY AND FOLIATIONS OF KNOT COMPLEMENTS

1993 ◽  
Vol 02 (04) ◽  
pp. 369-397 ◽  
Author(s):  
JOHN CANTWELL ◽  
LAWRENCE CONLON
Keyword(s):  
Unit Ball ◽  
Lattice Points ◽  
Compact Leaf ◽  
Minimal Genus ◽  
Relative Homology ◽  
Topological Description ◽  
Seifert Surfaces ◽  
Knot Complements ◽  
Interesting Class

An interesting class of knots have complement with a remarkably simple topological description. This class includes all the arborescent knots with only even weights hence, in particular, the two bridge knots and many knots of ten or fewer crossings. For these knots, there are choices of minimal genus Seifert surfaces S such that all taut, depth one foliations of the knot complement, having S as sole compact leaf, can be classified up to isotopy. These foliations correspond exactly to the lattice points over the open faces of the unit ball in a Thurston-like norm on the relative homology of the complement of S.

FIBERED TORTI-RATIONAL KNOTS

2010 ◽  
Vol 19 (10) ◽  
pp. 1291-1353 ◽  
Author(s):  
MIKAMI HIRASAWA ◽  
KUNIO MURASUGI
Keyword(s):  
Arbitrary Number ◽  
Continued Fraction ◽  
Alexander Polynomials ◽  
Minimal Genus ◽  
Seifert Surfaces ◽  
Tunnel Number ◽  
Rational Knots ◽  
Geometric Techniques ◽  
Continued Fraction Expansions ◽  
Satellite Knots

A torti-rational knot, denoted by K(2α, β|r), is a knot obtained from the 2-bridge link B(2α, β) by applying Dehn twists an arbitrary number of times, r, along one component of B(2α, β). We determine the genus of K(2α, β|r) and solve a question of when K(2α, β|r) is fibered. In most cases, the Alexander polynomials determine the genus and fiberedness of these knots. We develop both algebraic and geometric techniques to describe the genus and fiberedness by means of continued fraction expansions of β/2α. Then, we explicitly construct minimal genus Seifert surfaces. As an application, we solve the same question for the satellite knots of tunnel number one.

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.

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.

Finding Disjoint Seifert Surfaces

1988 ◽  
Vol 20 (1) ◽  
pp. 61-64 ◽  
Author(s):  
Martin Scharlemann ◽  
Abigail Thompson
Keyword(s):  
Seifert Surfaces

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.

THE MINIMAL GENUS PROBLEM IN RULED MANIFOLDS

2008 ◽  
Vol 17 (04) ◽  
pp. 471-482
Author(s):  
XU-AN ZHAO ◽  
HONGZHU GAO
Keyword(s):  
Cohomology Class ◽  
Standard Form ◽  
Geometric Construction ◽  
Diffeomorphism Group ◽  
Minimal Genus ◽  
Adjunction Formula ◽  
Embedded Surfaces

In this paper, we consider the minimal genus problem in a ruled 4-manifold M. There are three key ingredients in the studying, the action of diffeomorphism group of M on H2(M,Z), the geometric construction of surfaces representing a cohomology class and the generalized adjunction formula. At first, we discuss the standard form (see Definition 1.1) of a class under the action of diffeomorphism group on H2(M,Z), we prove the uniqueness of the standard form. Then we construct some embedded surfaces representing the standard forms of some positive classes, the generalized adjunction formula is used to show that these surfaces realize the minimal genera.

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