Toric code lattice drawing

Question

I am writing my thesis on the toric code with Overleaf, and for that I would like to draw:

1. a figure of the lattice with an example of the plaquette and vertice operators
2. a figure of the lattice with an example of the string operators
3. a figure of the torus with the grid drawn on the surface of the torus

I've been doing some research and this How to plot a lattice of points on the surface of a torus? question has been very helpful for figure 3), yet I don't know how to change the arrows on the surface for lines to create the grid. I copy the code and the difference I would like to make:

\documentclass[twocolumn]{revtex4}
\usepackage[inline]{asymptote}
\begin{document}
\begin{figure}[h]
\centering
\begin{asy}[width=5cm,height=5cm]
size(200);
import graph3;
pen surfPen=rgb(1,0.7,0);
pen xarcPen=deepblue+0.7bp;
pen yarcPen=deepred+0.7bp;
currentprojection=perspective(5,4,4);
real R=2;
real a=1;
triple fs(pair t) {
  return ((R+aCos(t.y))Cos(t.x),(R+aCos(t.y))Sin(t.x),a*Sin(t.y));
}
surface s=surface(fs,(0,0),(360,360),8,8,Spline);
draw(s,surfPen,render(compression=Low,merge=true));
int m=20;
int n=10;
real arcFactor=0.85;
pair p,q;
for(int i=1;i<=n;++i){
  for(int j=0;j<m;++j){
    p=(j360/m,(i%n)360/n);
    q=(((j+arcFactor)%m)360/m,i360/n);
    draw(fs(p)..fs((p+q)/2)..fs(q),xarcPen,Arrow3(size=4));
    q=(j360/m,((i%n)-arcFactor)360/n);
    draw(fs(p)..fs((p+q)/2)..fs(q),yarcPen,Arrow3(size=3));
    dot(fs(p));
  }
}
\end{asy}
\end{figure}
\end{document}

So really I just want to change "Arrow3" by some command that changes the arrows for lines.

Regarding figure 1) and 2), and for those not familiar with the toric code, this picture from article https://www.physics.rutgers.edu/grad/602/Lectures/JC_Presentations/0419/Intro_Toric_Code.pdf is a perfect example:

So a figure like this would be great (although obviously I don't want to copy it).

I don't mind if it's done with tikz, asymptote or any other application that might be useful. Any help or documentation that could be helpful? Thanks a lot.