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Plane circle immersions of rotation number k

Example

For every integer $k \neq 0$ the map $γ_{k} : S^{1} \to R^{2}$ , $γ_{k} (θ) = (\cos k θ, \sin k θ)$ , with $S^{1} = R / 2 π Z$ positively oriented, is an immersion: $γ_{k}^{'} (θ) = k (- \sin k θ, \cos k θ)$ never vanishes. Its unit tangent is $sgn (k) (- \sin k θ, \cos k θ)$ , a constant rotation of $e^{i k θ}$ , so $rot (γ_{k}) = k$ . The zero value is realised by $δ (θ) = (\cos θ, \sin 2 θ)$ , whose velocity is $p (\sin θ)$ for $p (u) = (- u, 2 - 4 u^{2})$ . This velocity never vanishes and contracts through nonzero loops to $(0, 2)$ via $p ((1 - t) \sin θ)$ , so $rot (δ) = 0$ . Thus ${γ_{k} : k \in Z ∖ {0}} \cup {δ}$ contains one representative of each regular homotopy class, by Whitney–Graustein under its inherited countable-choice hypothesis. The formula $γ_{0}$ is constant and is excluded.

Facts & Assumptions

Given: The oriented circle $S^{1} = R / 2 π Z$ , the maps $γ_{k} (θ) = (\cos k θ, \sin k θ)$ for $k \in Z ∖ {0}$ and $δ (θ) = (\cos θ, \sin 2 θ)$ .

[F1]

An immersion of the circle is a smooth map with everywhere nonvanishing velocity; its rotation number is the degree of the normalised velocity and equals the winding number of the velocity about the origin. Immersions, submersions, and constant-rank maps, Rotation number of an immersed oriented circle in the plane

[F2]

The degree of the $k$ -th power map of the circle is $k$ , and the winding number of a closed loop in $C^{\times}$ about $0$ is the degree of its normalised circle loop. Degree of the power map on the circle, For loops in C times, the winding number about 0 equals the circle degree, The degree of a based circle loop

[F3]

Two oriented plane circle immersions are regularly homotopic if and only if their rotation numbers agree, and the rotation number gives a bijection $π_{0} Imm (S^{1}, R^{2}) ≅ Z$ . Whitney–Graustein classification of plane circle immersions

Verification

1.1F1F2

For $k \neq 0$ , $γ_{k}$ has velocity $k (- \sin k θ, \cos k θ)$ of norm $∣ k ∣ > 0$ . Its normalized velocity is $sgn (k) i e^{i k θ}$ , a constant rotation of the degree- $k$ power map, so $rot (γ_{k}) = k$ . For negative $k$ the extra factor is $- 1$ ; it preserves degree.

1.2F1F2constructalgebra

The velocity of $δ$ is $p (\sin θ)$ with $p (u) = (- u, 2 - 4 u^{2})$ . This never vanishes, and $p ((1 - t) \sin θ)$ contracts it to $(0, 2)$ through nonzero loops, giving rotation number zero. If $δ (θ) = δ (ϕ)$ , equality of cosines gives $ϕ = θ$ or $ϕ = - θ$ modulo $2 π$ . In the second case equality of $\sin 2 θ$ and $- \sin 2 θ$ requires $\sin 2 θ = 0$ ; the only distinct pair is $π / 2, 3 π / 2$ , both mapping to the origin. Thus this is its unique double point.

2.1F1step 1.1

For $k < 0$ , $γ_{k} (θ) = γ_{∣ k ∣} (- θ)$ is the $∣ k ∣$ -fold circle with reversed domain orientation, consistent with its rotation number $k$ in step 1.1.

3.1F3step 1.1step 1.2step 2.1∎

By [F3], under its countable-choice hypothesis, $γ_{k}$ and $γ_{l}$ for nonzero $k, l$ are regularly homotopic exactly when $k = l$ , and no $γ_{k}$ is regularly homotopic to $δ$ . Steps 1.1 and 1.2 realise every integer with exactly one member of the displayed family. In particular rotation number zero does not force injectivity.

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Sources

Hassler Whitney, On regular closed curves in the plane, Compositio Mathematica 4 (1937), pp. 276–284 (standard reference, not scraped)
John Francis, The h-Principle, Lecture 10: Classifying immersions of spheres, after Smale (notes by A. Beaudry) (standard reference, not scraped)