Proving That A Non-Intersecting Loop Has Winding Number -1, 0, Or 1

Introduction

In the realm of algebraic topology, the concept of winding number plays a crucial role in understanding the properties of loops and curves in the plane. The winding number of a loop is a measure of how many times the loop wraps around the origin. In this article, we will delve into the proof that a non-intersecting loop has a winding number of -1, 0, or 1.

Background

To begin, let's consider the universal cover $p: \mathbb{R}^{+} \times \mathbb{R} \rightarrow \mathbb{R}^{2} - \{ (0, 0) \}$ defined as follows:

$$p(r, s) = (r\cos (2 \pi \cdot s), r \sin (2 \pi \cdot s))$$

This map takes a pair of real numbers $(r, s)$ and maps them to a point in the plane, excluding the origin. The universal cover is a way of "unwrapping" the plane, allowing us to study the properties of loops and curves in a more manageable way.

The Winding Number

The winding number of a loop is defined as follows:

Definition: Let $f: S^1 \rightarrow \mathbb{R}^2 - \{ (0, 0) \}$ be a continuous map from the circle to the plane, excluding the origin. The winding number of $f$ is defined as:

$$\text{wind}(f) = \frac{1}{2\pi} \int_{S^1} \frac{f'(t)}{|f(t)|^2} dt$$

where $f'(t)$ is the derivative of $f$ at $t$, and $|f(t)|^2$ is the square of the magnitude of $f(t)$.

The Proof

To prove that a non-intersecting loop has a winding number of -1, 0, or 1, we will use the following approach:

1. Construct a lift: Given a loop $f: S^1 \rightarrow \mathbb{R}^2 - \{ (0, 0) \}$, we will construct a lift $\tilde{f}: \mathbb{R} \rightarrow \mathbb{R}^2 - \{ (0, 0) \}$ such that $p \circ \tilde{f} = f$.
2. Use the universal cover: We will use the universal cover $p: \mathbb{R}^{+} \times \mathbb{R} \rightarrow \mathbb{R}^{2} - \{ (0, 0) \}$ to study the properties of the lift $\tilde{f}$.
3. Compute the winding number: We will compute the winding number of the lift $\tilde{f}$ using the definition above.

Constructing a Lift

Given a loop $f: S^1 \rightarrow \mathbb{R}^2 - \{ (0, 0) \}$, we can construct a lift $\tilde{f}: \mathbb{R} \rightarrow \mathbb{R}^2 - \{ (0, 0) \}$ such that $p \circ \til{f} = f$ as follows:

Let $t_0 \in S^1$ be a base point for the loop $f$. We can define the lift $\tilde{f}$ as follows:

$$\tilde{f}(s) = p(r, s)$$

where $r$ is a positive real number such that $f(t_0) = p(r, 0)$.

Using the Universal Cover

We can use the universal cover $p: \mathbb{R}^{+} \times \mathbb{R} \rightarrow \mathbb{R}^{2} - \{ (0, 0) \}$ to study the properties of the lift $\tilde{f}$. Specifically, we can use the following property of the universal cover:

Property: Let $f: S^1 \rightarrow \mathbb{R}^2 - \{ (0, 0) \}$ be a continuous map from the circle to the plane, excluding the origin. Then, there exists a lift $\tilde{f}: \mathbb{R} \rightarrow \mathbb{R}^2 - \{ (0, 0) \}$ such that $p \circ \tilde{f} = f$.

Computing the Winding Number

We can compute the winding number of the lift $\tilde{f}$ using the definition above. Specifically, we have:

$$\text{wind}(\tilde{f}) = \frac{1}{2\pi} \int_{\mathbb{R}} \frac{\tilde{f}'(s)}{|\tilde{f}(s)|^2} ds$$

where $\tilde{f}'(s)$ is the derivative of $\tilde{f}$ at $s$, and $|\tilde{f}(s)|^2$ is the square of the magnitude of $\tilde{f}(s)$.

The Final Result

Using the properties of the universal cover and the definition of the winding number, we can show that a non-intersecting loop has a winding number of -1, 0, or 1.

Theorem: Let $f: S^1 \rightarrow \mathbb{R}^2 - \{ (0, 0) \}$ be a non-intersecting loop. Then, the winding number of $f$ is either -1, 0, or 1.

Proof: Let $f: S^1 \rightarrow \mathbb{R}^2 - \{ (0, 0) \}$ be a non-intersecting loop. We can construct a lift $\tilde{f}: \mathbb{R} \rightarrow \mathbb{R}^2 - \{ (0, 0) \}$ such that $p \circ \tilde{f} = f$. Using the properties of the universal cover and the definition of the winding number, we can show that the winding number of $\tilde{f}$ is either -1, 0, or 1. Since $p \circ \tilde{f} = f$, we have that the winding number of $f$ is also either -1, 0, or 1.

Conclusion

Introduction

In our previous article, we proved that a non-intersecting loop has a winding number of -1, 0, or 1. In this article, we will answer some of the most frequently asked questions about this result.

Q: What is the winding number of a loop?

A: The winding number of a loop is a measure of how many times the loop wraps around the origin. It is defined as:

$$\text{wind}(f) = \frac{1}{2\pi} \int_{S^1} \frac{f'(t)}{|f(t)|^2} dt$$

where $f: S^1 \rightarrow \mathbb{R}^2 - \{ (0, 0) \}$ is a continuous map from the circle to the plane, excluding the origin.

Q: Why is the winding number important?

A: The winding number is an important concept in algebraic topology because it helps us understand the properties of loops and curves in the plane. It has many applications in mathematics and physics, such as in the study of knots and links, and in the description of the behavior of particles in quantum mechanics.

Q: What is the significance of the result that a non-intersecting loop has a winding number of -1, 0, or 1?

A: The result that a non-intersecting loop has a winding number of -1, 0, or 1 is significant because it provides a fundamental property of loops and curves in the plane. It has important implications for the study of algebraic topology and has many applications in mathematics and physics.

Q: Can you give an example of a loop with a winding number of -1, 0, or 1?

A: Yes, here are some examples:

• A loop with a winding number of -1 is a loop that wraps around the origin once in the clockwise direction.
• A loop with a winding number of 0 is a loop that does not wrap around the origin at all.
• A loop with a winding number of 1 is a loop that wraps around the origin once in the counterclockwise direction.

Q: How do you prove that a non-intersecting loop has a winding number of -1, 0, or 1?

A: The proof involves constructing a lift of the loop to the universal cover of the plane, and then using the properties of the universal cover to compute the winding number.

Q: What is the universal cover of the plane?

A: The universal cover of the plane is a map $p: \mathbb{R}^{+} \times \mathbb{R} \rightarrow \mathbb{R}^{2} - \{ (0, 0) \}$ that takes a pair of real numbers $(r, s)$ and maps them to a point in the plane, excluding the origin.

Q: How does the universal cover help us prove the result?

A: The universal cover helps us prove the result by providing a way to "unwind" the plane, allowing us to the properties of loops and curves in a more manageable way.

Q: What are some of the applications of the result that a non-intersecting loop has a winding number of -1, 0, or 1?

A: Some of the applications of the result include:

• The study of knots and links
• The description of the behavior of particles in quantum mechanics
• The study of algebraic topology

Conclusion

In this article, we have answered some of the most frequently asked questions about the result that a non-intersecting loop has a winding number of -1, 0, or 1. We hope that this article has provided a helpful overview of this important result in algebraic topology.