Closed curves of constant torsion. II

The Bäcklund transformation for pseudospherical surfaces, which is equivalent to that of the sine-Gordon equation, can be restricted to give a transformation on space curves that preserves constant torsion. We study its effects on closed curves (in particular, elastic rods) that generate multiphase solutions for the vortex filament flow (also known as the Localized Induction Equation). In doing so, we obtain analytic constant-torsion representatives for a large number of knot types.

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Generating the Mapping Class Group

10.23943/princeton/9780691147949.003.0005 ◽

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.

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

Ukrainian Mathematical Bulletin ◽

10.37069/1810-3200-2020-17-2-7 ◽

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.

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