Algebraic condition for a twist in 2D ball or a hypersphere? We define twisted ball here as a ball where only one point (the twist) separates the ball sides.
Simple implicit presentation for the 2D Ball is $x^2+y^2=r^2$. I am trying to find a general condition when the 2D ball has a crossing like the last parametric plot. Third one is just a line so it can also be considered as a naive twisted ball: is the twisted ball just a line with certain features? The extra 2 in the fourth plot results into a more apparent twisted ball: is there some other feature we should require for a twisted ball?

Questions


How to deduce the algebraic condition for the 2D twisted ball?

Easier to consider higher dimensional balls also, hyperspheres?

Please explain the twist in the ball.
Let us consider firstly the set of parametric plots in which the coordinates $x,y$ are given by $ [\sin(t), \cos(kt)]$. I will restrict my answer to the cases where $k$ is an integer.

In this set of parametric plots, the function crosses the $x$-axis for $2k$ different values of $t$, which are  those satisfying  $t=\pi(j-1/2)/k$, where $j=1,2,3...2k$. However, because of the symmetry of these $t$ values with respect to the $y$-axis, these zeros can be grouped in pairs with the same $x$-value, i.e. corresponding to the same point on the $x$-axis. As a result, the function crosses the $x$-axis in $k$ points when $k$ is even, and $k+1$ points when $k$ is odd.

For example, when $k=4$, the function crosses the $x$-axis for eight values of $t$, namely $\frac{\pi}{8}, \frac{3}{8}\pi, \frac{5}{8}\pi... \frac{15}{8}\pi$. However, these eight values correspond to only four points on the $x$-axis, since the $x$ value is equal for the pair $t=\frac{\pi}{8}$ and $t=\frac{7}{8}\pi$, for the pair $t=\frac{3}{8}\pi$ and $t=\frac{5}{8}\pi$, and so on. Accordingly, the parametric plot $ [\sin(t), \cos(4t)]$ is as follows:



On the other hand, for example, when $k=3$, the function crosses the $x$-axis for six values of $t$, namely $\frac{\pi}{6}, \frac{3}{6}\pi, \frac{5}{6}\pi... \frac{11}{6}\pi$. These six values correspond to only four points on the $x$-axis, since the $x$ value is equal for the pair $t=\frac{\pi}{6}$ and $t=\frac{5}{6}\pi$, and for the pair $t=\frac{7}{6}\pi$ and $t=\frac{11}{6}\pi$; the other two points are given by $t=\frac{3}{6}\pi$ and $t=\frac{9}{6}\pi$.  Accordingly, the parametric plot $ [\sin(t), \cos(3t)]$ is as follows:



Note then that, when $k$ is even, all $2k$ zeros of the function can be grouped in pairs, leading to $k$ points on the $x$-axis; conversely, when $k$ is odd, all $2k$ zeros of the function can be grouped in pairs, except two that are "unpaired" (those corresponding to $t=\frac{\pi}{2}$ and $t=\frac{3}{2}\pi$). This leads to $k+1$ points on the $x$-axis.  Also note other two things: 1) for $k$ odd, the two "unpaired" zeros given by $t=\frac{\pi}{2}$ and $t=\frac{3}{2}\pi$ correspond to the most lateral points (left and right) of the function; 2) for $k=1$, the function crosses the $x$-axis only in these two unpaired zeros, thus without crossing itself (in fact for $k=1$ it reduces to a circle).

Now it is also not difficult to show that, considering the behaviour of the function in these points laying on the $x$-axis, its derivative has a unique value when $k$ is even - which means that the function does not cross itself. On the other hand, when $k$ is odd, the derivative assumes two different values (positive and negative) in each of these points laying on the $x$-axis except for the two "unpaired", most lateral ones. This means that the function crosses itself in $k-1$ points on the $x$-axis. For example, we expect that for $k=14$ the parametric plot $ [\sin(t), \cos(kt)]$ crosses the $x$-axis in $14$ points and does not cross itself. This is demonstrated by the following figure showing $ [\sin(t), \cos(14t)]$:

 

On the other hand, for example we expect that for $k=9$ the parametric plot $ [\sin(t), \cos(kt)]$ crosses the $x$-axis in $9+1=10$ points, and that it crosses itself in $9-1=8$ of these points. This is demonstrated by the following plot of $ [\sin(t), \cos(9t)]$, showing the $8$ points where the function crosses itself and the remaining $2$ most lateral points ("unpaired" zeros) where the function crosses the $x$-axis:



As a result, within the set of parametric plots $x,y$ given by $ [\sin(t), \cos(kt)]$, there is no possibility to find a case that yields a unique crossing point dividing the function in two sides (left and right). For $k$ even, the function does not cross itself; among the odd $k$ values, for $k=1$ the function does not cross itself; for $k=3$, the function crosses the x-axis in four points (two of which are the lateral, "unpaired" ones), and therefore crosses itself in two points on the $x$-axis; lastly, as already shown, for higher odd values of $k$, the function crosses itself on the $x$-axis even more times. 

All these considerations can be repeated for the set of parametric plots in which the coordinates $x,y$ are given by $ [\sin(t), \sin(kt)]$, with the only difference that the behaviour of the function for $k$ odd or even is inverted. More precisely, in this case the function crosses the $x$-axis $k$ times and never crosses itself when $k$ is odd; conversely, the function crosses the $x$-axis in $k+1$ points and crosses itself in $k-1$ of these points when $k$ is even. 

As a result, within the set of parametric plots $x,y$ given by $ [\sin(t), \sin(kt)]$, the only case in which there is a unique crossing point dividing the function in two sides (left and right) occurs for $k=2$. In fact, in this case, the function crosses the $x$-axis in three points (two of which are the"unpaired" lateral ones) and crosses itself in a single central point, yielding a "twisted" ball according to the definition given in the OP. Accordingly, the plot of $ [\sin(t), \sin(2t)]$ is as follows:
I have some sort of intuitive touch on these twists based on Quantum physics - they like trigger sudden peaks which then again change the architecture of the object through Gibbs' phenomenon.
So the algebraic condition would be any sudden peak in a 2D twisted ball.