Regular homotopy for immersions of graphs into surfaces

2016 ◽  
Vol 207 (6) ◽  
pp. 854-872 ◽  
Author(s):  
D A Permyakov
Keyword(s):  
Regular Homotopy

Immersions with non-zero normal vector fields

1992 ◽  
Vol 112 (2) ◽  
pp. 281-285 ◽  
Author(s):  
Bang-He Li ◽  
Gui-Song Li
Keyword(s):  
Vector Fields ◽  
Normal Vector ◽  
Homotopy Classes ◽  
Natural Action ◽  
Singularity Method ◽  
One To One ◽  
Regular Homotopy

Let M be a smooth n-manifold, X be a smooth (2n − 1)-manifold, and g:M → X be a map. It was proved in [6] that g is always homotopic to an immersion. The set of homotopy classes of monomorphisms from TM into g*TX, which is denoted by Sg, may be enumerated either by the method of I. M. James and E. Thomas or by the singularity method of U. Koschorke (see [1] and references therein). When the natural action of π1(XM, g) on Sg is trivial, for example, if X is euclidean, the set Sg is in one-to-one correspondence with the set of regular homotopy classes of immersions homotopic to g (see e.g. [4]).

Basic Geometric Constructions

2021 ◽  
pp. 217-226
Author(s):  
Mark Powell ◽  
Arunima Ray
Keyword(s):  
Dimensional Space ◽  
Intersection Point ◽  
Important Application ◽  
Geometric Constructions ◽  
Clifford Torus ◽  
The Cost ◽  
Regular Homotopy ◽  
Associated Surfaces

Basic geometric constructions, including tubing, boundary twisting, pushing down intersections, and contraction followed by push-off are presented. These moves are used repeatedly later in the proof. New, detailed pictures illustrating these constructions are provided. The Clifford torus at an intersection point between two surfaces in 4-dimensional space is described. The chapter closes with an important application of some of these moves called the Geometric Casson Lemma. This lemma upgrades algebraically dual spheres to geometrically dual spheres, at the cost of introducing more self-intersections. It is also shown that an immersed Whitney move is a regular homotopy of the associated surfaces.

scholarly journals Tight surfaces and regular homotopy

Topology ◽  
1986 ◽  
Vol 25 (4) ◽  
pp. 475-481 ◽  
Author(s):  
U. Pinkall
Keyword(s):  
Regular Homotopy

scholarly journals Gamma-polynomial and its generalization to a 2-string tangle polynomial

2015 ◽  
Vol 24 (08) ◽  
pp. 1550047 ◽  
Author(s):  
Tomoko Yanagimoto
Keyword(s):  
Existence Proof ◽  
Polynomial Invariant ◽  
Regular Homotopy ◽  
Knot Diagrams ◽  
Knot Polynomial

The zeroth coefficient polynomial of the skein (HOMFLYPT) knot polynomial called the Γ-polynomial is studied from a viewpoint of regular homotopy of knot diagrams. In particular, an elementary existence proof of the knot invariance of the Γ-polynomial is given. After observing that there are three types for 2-string tangle diagrams, the Γ-polynomial is generalized to a polynomial invariant of a 2-string tangle. As an application, we have a new proof of the assertion that Kinoshita's θ-curve is not equivalent to the trivial θ-curve.

On Regular Homotopy in Codimension 1

1966 ◽  
Vol 83 (2) ◽  
pp. 257 ◽  
Author(s):  
Valentin Poenaru
Keyword(s):  
Regular Homotopy

scholarly journals REGULAR HOMOTOPIC DEFORMATION OF COMPACT SURFACE WITH BOUNDARY AND MAPPING CLASS GROUP

2011 ◽  
Vol 20 (10) ◽  
pp. 1391-1396
Author(s):  
SUSUMU HIROSE ◽  
AKIRA YASUHARA
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Compact Surface ◽  
Mapping Class ◽  
Algebraic Condition ◽  
Necessary And Sufficient ◽  
Regular Homotopy

A necessary and sufficient algebraic condition for a diffeomorphism over a surface embedded in S3 to be induced by a regular homotopic deformation is discussed, and a formula for the number of signed pass moves needed for this regular homotopy is given.

scholarly journals A frame energy for immersed tori and applications to regular homotopy classes

2016 ◽  
Vol 104 (1) ◽  
pp. 143-186
Author(s):  
Andrea Mondino ◽  
Tristan Rivière
Keyword(s):  
Homotopy Classes ◽  
Regular Homotopy
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