The winding number of a closed curve around a point

Abstract

In this expository note we present an elementary direct rigorous definition and the simplest properties of the winding number. This definition is simpler than the one given in some textbooks. We show how to compute the winding number easily: using additivity or counting the (signed) intersection points. In the language of the winding number, we present an elementary formulation and proof of the low-dimensional case of the Borsuk--Ulam theorem. An English version is followed by a Russian version.

Figures (7)

• Figure 2.1: $w(ABC) = +1$ и $w(ABCD)= 0$
• Figure 2.2: Числа оборотов равны $0,~+1,~-1,~+2$
• Figure 3.1: Замкнутая ломаная $l$, симметричная относительно точки $O$; $w(l)=3$
• Figure 4.1: Шахматные раскраски дополнений и внутренности по модулю 2
• Figure 4.2: Знак точки пересечения

Theorems & Definitions (11)

• proof : Доказательство пункта (a)
• proof : Построение
• proof
• proof
• proof : Доказательство теоремы \ref{['borul-pl']}
• proof
• proof
• proof
• proof : Набросок доказательства
• proof : Набросок доказательства