scholarly journals Minimal genus problem for T2–bundles over surfaces

2021 ◽  
Vol 21 (2) ◽  
pp. 893-916
Author(s):  
Reito Nakashima
Keyword(s):  
Minimal Genus

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.

scholarly journals Minimal genus of a multiple and Frobenius number of a quotient of a numerical semigroup

2015 ◽  
Vol 25 (06) ◽  
pp. 1043-1053 ◽  
Author(s):  
Francesco Strazzanti
Keyword(s):  
Positive Integer ◽  
Numerical Semigroup ◽  
Frobenius Number ◽  
Numerical Semigroups ◽  
Minimal Genus

Given two numerical semigroups S and T and a positive integer d, S is said to be one over d of T if S = {s ∈ ℕ | ds ∈ T} and in this case T is called a d-fold of S. We prove that the minimal genus of the d-folds of S is [Formula: see text], where g and f denote the genus and the Frobenius number of S. The case d = 2 is a problem proposed by Robles-Pérez, Rosales, and Vasco. Furthermore, we find the minimal genus of the symmetric doubles of S and study the particular case when S is almost symmetric. Finally, we study the Frobenius number of the quotient of some families of numerical semigroups.

scholarly journals The minimal genus problem for elliptic surfaces

2014 ◽  
Vol 200 (1) ◽  
pp. 127-140
Author(s):  
M. J. D. Hamilton
Keyword(s):  
Elliptic Surfaces ◽  
Minimal Genus

scholarly journals Incidence Matrices, Permutation Characters, and the Minimal Genus of a Permutation Group

2002 ◽  
Vol 98 (1) ◽  
pp. 87-105 ◽  
Author(s):  
Daniel Frohardt ◽  
Robert Guralnick ◽  
Kay Magaard
Keyword(s):  
Permutation Group ◽  
Incidence Matrices ◽  
Minimal Genus

scholarly journals GEOMETRY OF THE HOMOLOGY CURVE COMPLEX

2012 ◽  
Vol 04 (03) ◽  
pp. 335-359 ◽  
Author(s):  
INGRID IRMER
Keyword(s):  
Homotopy Class ◽  
Intersection Number ◽  
Simple Algorithm ◽  
Curve Complex ◽  
Oriented Surface ◽  
Immersed Surfaces ◽  
Minimal Genus ◽  
Closed Oriented Surface

Suppose S is a closed, oriented surface of genus at least two. This paper investigates the geometry of the homology multicurve complex, [Formula: see text], of S; a complex closely related to complexes studied by Bestvina–Bux–Margalit and Hatcher. A path in [Formula: see text] corresponds to a homotopy class of immersed surfaces in S × I. This observation is used to devise a simple algorithm for constructing quasi-geodesics connecting any two vertices in [Formula: see text], and for constructing minimal genus surfaces in S × I. It is proven that for g ≥ 3 the best possible bound on the distance between two vertices in [Formula: see text] depends linearly on their intersection number, in contrast to the logarithmic bound obtained in the complex of curves. For g ≥ 4 it is shown that [Formula: see text] is not δ-hyperbolic.

scholarly journals Minimal genus embeddings in $S^2-$bundles over surfaces

1997 ◽  
Vol 4 (3) ◽  
pp. 379-394 ◽  
Author(s):  
Bang-He Li ◽  
Tian-Jun Li
Keyword(s):  
Minimal Genus

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.

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