How to draw this picture (draw arrows in middle of a loop generated by 'control', make break ) using tikz

Question

I would like to draw this picture by [tikz]. The elliptic and circle are easy, and I use 'control' to draw the red loop. But I do not know how to draw the arrow in the middle of the loop and do not know how to make spaces in the intersections.

Here is my codes.

\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
\draw (0,0) ellipse (20pt and 10pt);
\draw (1,0) ellipse (20pt and 20pt);
\filldraw [red] (1,-1.5) circle (1pt);
\draw[draw=red] (1,-1.5) .. controls (0.2,0.2) and (1.8,0.2).. (1,-1.5);
\draw[draw=red] (1,-1.5) .. controls (-0.5,0.6) and (-2,0.2) .. (1,-1.5);
\end{tikzpicture}
\end{document}

And here is what I got

I try to use 'decoration', but it does not work well for 'control'.

Answer 1

Here's an example using the knots library for the intersections and decorations.markings for the arrows.

\documentclass{article}
%\url{https://tex.stackexchange.com/q/621085/86}
\usepackage{tikz}
\usetikzlibrary{
  knots,
  decorations.markings,
  arrows.meta
}
\begin{document}
\begin{tikzpicture}
\begin{knot}[
%  draft mode=crossings,
  flip crossing/.list={6,4,1},
  end tolerance=12pt,
]
\strand (0,0) ellipse[x radius=20pt, y radius=10pt];
\strand (1,0) circle[radius=20pt];
\strand[
  draw=red,
  only when rendering/.style={
    postaction=decorate,
  },
  decoration={
    markings,
    mark=at position .47 with {\arrowreversed{To}}
  }
] (1,-1.5) .. controls (0.2,0.2) and (1.8,0.2).. (1,-1.5);
\strand[
  draw=red,
  only when rendering/.style={
    postaction=decorate,
  },
  decoration={
    markings,
    mark=at position .45 with {\arrow{To}}
  }
] (1,-1.5) .. controls (-0.5,0.6) and (-2,0.2) .. (1,-1.5);
\end{knot}
\filldraw [red] (1,-1.5) circle[radius=1pt];
\end{tikzpicture}
\end{document}

Answer 2

Drawing arrows in the middle of a line is addressed here. In the code below I defined the style midarr to place the arrow.

The knots library can be used as in Andrew Stacy's solution, but if you want to do this manually, you can use double=<linecolor> and double distance=<linethickness>. I defined a style overcross that will produce the desired effect:

\tikzset{overcross/.style={double, line width=1.5, white, double=#1, double distance=\knotwidth},
overcross/.default=black}

The macro \knotwidth can be set to the desired line thickness for your diagrams. You can adjust the spacing around the overcrossing by changing the 1.5.

To make this work, draw the under-segments first. Then use \draw[overcross] or \draw[overcross, double=red] for the overcrossings. Note that this means splitting up segments into pieces. I set the in and out angles to make them look smooth.

Here is the code:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}
\tikzset{overcross/.style={double, line width=1.5, white, double=#1, double distance=\knotwidth},
    overcross/.default=black}
\tikzset{midarr/.style={decoration={markings,mark={at position .5 with {\arrow{>}}}},postaction={decorate}}}
\newcommand{\knotwidth}{.8pt}
\begin{document}
\begin{tikzpicture}[line width=\knotwidth]
\draw (0,2) to[out=90, in=90] (-2,2);
\draw[red] (0,1.5) to[out=180, in=105] (-.25,.75);
\draw[red, midarr] (0,0) to[out=65, in=-105] (.25,.75);
\draw[overcross, double=red] (-1.75,1.9) to[out=180, in=90] (-2.2,1.3);
\draw[overcross] (0,2) circle[radius=1];
\draw[overcross] (-2,2) to[out=-90, in=-90] (0,2);
\draw[overcross, double=red] (.25,.75) to[out=75, in=0] (0,1.5);
\draw[overcross, double=red] (0,0) to[out=180, in=0] (-1.75,1.9);
\draw[red, midarr] (-2.2,1.3) to[out=-90, in=180] (0,0);
\draw[red] (-.25,.75) to[out=-75, in=115] (0,0);
\fill[red] (0,0) circle[radius=.05];
\end{tikzpicture}
\end{document}

Answer 3

Here is an example where to draw the arrows I just drew each loop one half at a time, and to generate the intersections I conveniently ordered the drawing of each element.

\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
\coordinate (A) at (1.1,-1.4);
\coordinate (B) at (-0.2,-0.1);
\coordinate (C) at (1.1,-0.4);
\draw (1,0) + (180:20pt) arc (180:245:20pt);
\draw[red] (B) to[out=180,in=175] (A);
\draw[white,line width=2.5pt] (0.3,0) ellipse (20pt and 10pt);
\draw (0.3,0) ellipse (20pt and 10pt);
\draw[red] (C) to[out=180, in=130] (A);
\draw[white,line width=2.5pt] (1,0) + (160:20pt) arc (160:-115:20pt);
\draw (1,0) + (-115:20pt) arc (-115:180:20pt);
\draw[white,line width=2.5pt] (A) + (156:10pt)  to[out=135,in=0] (B);
\draw[->,red] (A) to[out=175,in=0] (B);
\draw[white,line width=2.5pt] (A) to[out=50,in=0] (C);
\draw[->,red] (A) to[out=50,in=0] (C);
\filldraw [red] (A) circle (1pt);
\end{tikzpicture}
\end{document}