For each of the following plane curves, either prove that it is a Jordan curve or prove that it fails to be a Jordan curve.

1. \(\gamma\colon [-2,2]\longrightarrow\mathbb{R}^2\) defined by \(\gamma(t)=(t^3-4,t^2-4).\)

2. \(\gamma\colon [0,2\pi]\longrightarrow\mathbb{R}^2\) defined by \(\gamma(t)=((2\operatorname{cos}(t)-1)\operatorname{cos}(t),(2\operatorname{cos}(t)-1)\operatorname{sin}(t)).\)

3. \(\gamma\colon [0,2\pi]\longrightarrow\mathbb{R}^2\) defined by \[\gamma(t) = \begin{cases} (\operatorname{cos}(t),\operatorname{sin}(t)) & t\in [0,\pi] \\ (\operatorname{cos}(t),0) & t\in[\pi,2\pi] \\ \end{cases}\]

Let \(\gamma : [a,b] \to \mathbb{R}^2\) be a smooth immersed closed curve with signed curvature \(\kappa : [a,b]\to \mathbb{R}.\) The integer \[R_{\gamma}=\frac{1}{2\pi}\int_{a}^b \kappa(t)\Vert\gamma'(t)\Vert\mathrm{d}t\] is called the rotation index of \(\gamma\). Note that this restricts to give Definition 2.36 in the case that \(\gamma\) is unit speed. Sketch the following curves and then compute their rotation index.

1. Fix \(n\in\mathbb{Z}\) and let \(\gamma_n\colon [0,2\pi]\longrightarrow \mathbb{R}^2\) be given by \(\gamma_n(t)=(\operatorname{cos}(nt),\operatorname{sin}(nt)).\)

2. \(\gamma\colon [0,2\pi]\longrightarrow \mathbb{R}^2\) be given by \(\gamma(t)=(\operatorname{cos}(t),\operatorname{sin}(2t)).\)

3. \(\gamma\colon [0,2\pi]\longrightarrow \mathbb{R}^2\) be given by \(\gamma(t)=(\operatorname{sin}(t),\operatorname{sin}(2t)).\)

4. \(\gamma\colon [0,2\pi]\longrightarrow\mathbb{R}^2\) defined by \(\gamma(t)=((2\operatorname{cos}(t)-1)\operatorname{cos}(t),(2\operatorname{cos}(t)-1)\operatorname{sin}(t)).\)

Let \(\gamma\colon I\longrightarrow\mathbb{R}^3\) be an arc length parametrized curve.

1. Prove that \(\kappa(t)= \Vert \ddot\gamma(t)\Vert.\)

2. Assume \(\kappa(t)\ne 0,\) \(\kappa'(t)\ne0,\) and \(\tau(t)\ne0,\) for all \(t\in I.\) Then \(\gamma(I)\) lies on a sphere if and only if the following equation holds for some constant \(c\in \mathbb{R},\) \[c=R^2+(R')^2Q^2,\] where \(R:=\frac{1}{\kappa}\) and \(Q:=\frac{1}{\tau}.\)