Some Characterizations of Arcs and Simple Closed Curves

1948 ◽  
Vol 70 (3) ◽  
pp. 497 ◽  
Author(s):  
R. H. Bing
Keyword(s):  
Closed Curves

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

2020 ◽  
Vol 17 (2) ◽  
pp. 256-277
Author(s):  
Ol'ga Veselovska ◽  
Veronika Dostoina
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Analytic Functions ◽  
Combinatorial Identities ◽  
Independent Interest ◽  
Closed Curves ◽  
In Series ◽  
Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

Nontransversal curves of T-points: a source of closed curves of global bifurcations

2002 ◽  
Vol 303 (2-3) ◽  
pp. 204-211 ◽  
Author(s):  
A. Algaba ◽  
F. Fernández-Sánchez ◽  
E. Freire ◽  
M. Merino ◽  
A.J. Rodrı́guez-Luis
Keyword(s):  
Global Bifurcations ◽  
Closed Curves

scholarly journals Stability of Reeb Graphs of Closed Curves

2012 ◽  
Vol 283 ◽  
pp. 71-76
Author(s):  
Barbara Di Fabio ◽  
Claudia Landi
Keyword(s):  
Closed Curves ◽  
Reeb Graphs

scholarly journals Topological Quantum Computing and 3-Manifolds

2021 ◽  
Vol 3 (1) ◽  
pp. 153-165
Author(s):  
Torsten Asselmeyer-Maluga
Keyword(s):  
Quantum Computing ◽  
Berry Phase ◽  
Basic Structure ◽  
Quantum States ◽  
Point Of View ◽  
Surface Topology ◽  
Usual Sense ◽  
Closed Curves ◽  
Topological Quantum ◽  
2D System

In this paper, we will present some ideas to use 3D topology for quantum computing. Topological quantum computing in the usual sense works with an encoding of information as knotted quantum states of topological phases of matter, thus being locked into topology to prevent decay. Today, the basic structure is a 2D system to realize anyons with braiding operations. From the topological point of view, we have to deal with surface topology. However, usual materials are 3D objects. Possible topologies for these objects can be more complex than surfaces. From the topological point of view, Thurston’s geometrization theorem gives the main description of 3-dimensional manifolds. Here, complements of knots do play a prominent role and are in principle the main parts to understand 3-manifold topology. For that purpose, we will construct a quantum system on the complements of a knot in the 3-sphere. The whole system depends strongly on the topology of this complement, which is determined by non-contractible, closed curves. Every curve gives a contribution to the quantum states by a phase (Berry phase). Therefore, the quantum states can be manipulated by using the knot group (fundamental group of the knot complement). The universality of these operations was already showed by M. Planat et al.

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