Drawing knots without overdrawing a grid

The latest (on CTAN and TL as of Feb 2021) version of the spath3 library includes new routines to work with paths, including splitting them at intersections and inserting gaps. This makes it possible to draw knot diagrams by a different means to the knots tikzlibrary and is particularly well suited to this situation where there the background is not uniform.

Here's the code:

documentclass{article}
%url{https://tex.stackexchange.com/q/388611/86}
 usepackage{tikz}
 usetikzlibrary{spath3,intersections}
 begin{document}

 begin{tikzpicture}[baseline]
 draw [lightgray] (0,0) grid (2,2);
 path[spath/save=strand] (0,0) -- (2,2) -- (2,0) -- (0,2);
tikzset{
  every strand component/.style={draw,thick},
  spath/knot={strand}{8pt}{2}
}
 end{tikzpicture}

 end{document}

And here's the result:

Here's a solution using transparency groups. It's not the most elegant solution as it requires drawing the knot path twice, but it works with the blue colour as you comment on Heiko's answer.

It works as following:

1. Draw the grid.
2. Draw the full knot path over the grid in white to erase the parts of the grid that the knot will go over.
3. Draw the knot but multiply the colours when rendering onto the grid.

Here's the code:

documentclass{article}
%url{https://tex.stackexchange.com/q/388611/86}
 usepackage{tikz}
usetikzlibrary{knots}

 begin{document}

begin{tikzpicture}[baseline]
begin{scope}[blend group=multiply]
begin{scope}[blend group=normal]
 draw [lightgray,ultra thin] (0,0) grid (2,2);
 draw [thick,white] (0,0) -- (2,2) -- (2,0) -- (0,2);
end{scope}
begin{scope}[blend group=normal]
begin{knot}[consider self intersections, clip width=10]
 strand[thick,blue] (0,0) -- (2,2) -- (2,0) -- (0,2);
 flipcrossings{2}
 end{knot}
end{scope}
end{scope}

end{tikzpicture}
 hspace{1cm} 
 begin{tikzpicture}[baseline]
 draw [thick] (2,0) -- (0,2);
 draw [white, line width=2ex] (0,0) -- (2,2);
 draw [lightgray] (0,0) grid (2,2);
 draw [thick] (0,0) -- (2,2) -- (2,0);
 end{tikzpicture}

 end{document}

Result (zooming in on the crossing):

It's not completely perfect as this screenshot shows - drawing the strand in white potentially erases a bit more than strictly necessary.

Since the grid is light gray and the knot lines are black, there is a trick with transparency. The grid is drawn on top of the knots, but with opacity=.25. Then, the black lines remain black and the grid lines on top of the white background become light gray:

documentclass{article}
usepackage{tikz}
usetikzlibrary{knots}
begin{document}
begin{tikzpicture}[baseline]
 begin{knot}[consider self intersections, clip width=10]
   strand [thick, line join=round]
     (0,0) -- (2,2) -- (2,0) -- (0,2);
   flipcrossings{2}
  end{knot}
  draw [opacity=.25, line cap=rect] (0, 0) grid (2,2);
end{tikzpicture}
end{document}

With white background:

I guess in your case the most simple way is to move the grid statement after drawing the knots.